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Few topics connect with as many branches of mathematics as conic sections. In this rich collection of activities, students build their own physical models of conic sections and then construct more flexible models with Sketchpad. And by developing both geometric and analytic proofs, students connect these mathematical realms in ways essential to any second-year algebra, precalculus, or analytic geometry course.

The sketches referenced in the book can be downloaded from https://sketchpad.keycurriculum.com/KeyModules/index.html.


Year:
2012
Publisher:
Key Curriculum Press
Language:
english
Pages:
96
ISBN 10:
1604402776
ISBN 13:
9781604402773
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with

DANIEL SCHER

Prepress and Printer:

Steven Chanan
Jennifer Strada
Caroline Ayres
Jason Luz
Joan Saunders
Debbie Cogan
Diana Jean Parks
Ariana Grabec–Dingman
Private Collection, Bridgeman Art
Library; Paul Eekhof, Masterfile
Lightning Source, Inc.

Executive Editor:
Publisher:

Casey FitzSimons
Steven Rasmussen

Project Editor:
Production Editor:
Art and Design Coordinator:
Art Editor:
Copyeditor:
Manager, Editorial Production:
Production Director:
Cover Designer:
Cover Photo Credits:

Cover art and text include illustrations from Exercitationum Mathematicorum, Liber IV,
Organica Conicarum Sectionum in Plano Descriptione by Frans van Schooten (Leiden, 1646).
An original copy of this book is housed at Cornell’s Kroch Library.
®Key Curriculum Press is a registered trademark of Key Curriculum Press.
®The Geometer’s Sketchpad and ®Dynamic Geometry are registered trademarks of
KCP Technologies. ⇥Sketchpad is a trademark of KCP Technologies. All other brand
names and product names are trademarks or registered trademarks of their respective
holders.

Limited Reproduction Permission
© 2012 Key Curriculum Press. All rights reserved. Key Curriculum Press grants the
teacher who purchases Exploring Conic Sections with The Geometer’s Sketchpad the right to
reproduce activities and example sketches for use with his or her own students.
Unauthorized copying of Exploring Conic Sections with The Geometer’s Sketchpad or of the
Exploring Conic Sections sketches is a violation of federal law.
Key Curriculum Press
1150 65th Street
Emeryville, California 94608
510-595-7000
editorial@keypress.com
www.keypress.com
10 9 8 7 6 5 4 3 2 1
ISBN 978-1-60440-277-3

15 14 13 12

Contents
Downloading Sketchpad Documents ........................................................ iv
Introduction ....................................................................................................... v
Customizing the Activities .............................................................................. vi
Acknowledgments .............; ............................................................................... vi
Common Commands and Shortcuts ....................................................... viii
Chapter 1: Ellipses
Chapter Overview................................................................................................ 3
Getting Started: When Is a Circle Not a Circle? ............................................. 5
The Pins-and-String Construction .................................................................... 6
The Concentric Circles Construction ............................................................... 8
Some Ellipse Relationships .............................................................................. 11
The Folded Circle Construction ...................................................................... 13
The Congruent Triangles Construction ......................................................... 17
The Carpenter’s Construction ......................................................................... 19
Danny’s Ellipse .................................................................................................. 25
Ellipse Projects ................................................................................................... 28
Chapter 2: Parabolas
Chapter Overview ............................................................................................. 33
Introducing the Parabola ................................................................................. 34
The Folded Rectangle Construction ............................................................... 38
The Expanding Circle Construction ............................................................... 42
Parabola Projects ............................................................................................... 45
Chapter 3: Hyperbolas
Chapter Overview ............................................................................................. 49
Introducing the Hyperbola .............................................................................. 50
The Concentric Circles Construction ............................................................. 52
Hyperbola Projects ............................................................................................ 55
Chapter 4: Optimization
Chapter Overview ............................................................................................. 59
The Burning Tent Problem .............................................................................. 60
The Swimming Pool Problem ......................................................................... 63
An Optimization Project .................................................................................. 66
Activity Notes
Chapter 1: Ellipses ............................................................................................ 69
Chapter 2: Parabolas ......................................................................................... 78
Chapter 3: Hyperbolas ..................................................................................... 82
Chapter 4: Optimization .................................................................................. 84
Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Contents • iii

Downloading Sketchpad Documents
All Sketchpad documents (sketches) for Exploring Conic Sections with
The Geometer’s Sketchpad are available online for download.
• Go to www.keypress.com/gspmodules
• Log in using your Key Online account, or create a new account and
log in.
• Enter this access code: ECSHS633
• A Download Files button will appear. Click to download a compressed
(.zip) folder of all sketches for this book.
The downloadable folder contains all of the sketches you need for this
book, organized by chapter and activity. The sketches require
The Geometer’s Sketchpad Version 5 software to open. Go to
www.keypress.com/gsp/order to purchase or upgrade Sketchpad, or
download a trial version from www.keypress.com/gsp/download.

iv • Downloading Sketchpad Documents

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Introduction
First, a confession: As a student, I didn’t place conic sections on my list
of favorite high-school topics.
The standard textbook treatment of the ellipse, parabola, and hyperbola
seemed uninspired. There were messy algebraic equations with multiple
square roots. There was lots of terminology. Drawing a conic meant
plotting several points on graph paper and connecting them with a
wobbly curve.
I gave little thought to conics until I met David Dennis, then a fellow
graduate student in mathematics education at Cornell University.
David had a keen interest in the origins of curve-drawing devices.
One historical figure in particular intrigued him: Frans van Schooten
(1615–1660), a Dutch mathematician whose translation and commentary
for Descartes’ landmark treatise, La Géométrie, led to the popularization of
Cartesian geometry.
A free online version
of van Schooten’s
book is available on
the website of
European Cultural
Heritage Online.

Descartes’ book was available in bookstores, but van Schooten’s work,
Exercitationum Mathematicorum, Liber IV, Organica Conicarum Sectionum in
Plano Descriptione (A Treatise on Devices for Drawing Conic Sections),
remained tucked away in rare book collections. Luckily, Cornell’s Kroch
Library contained a copy. David made the trip.
When he returned from the library, David could barely contain his
enthusiasm. “Go and see it for yourself,” he said. “It’s really something.”
A visit to the library confirmed David’s findings. There in van Schooten’s
book were drawing after drawing of devices that drew conics. The illustrations were exquisite, with artistic flourishes adorning many of them.
Viewing van Schooten’s curve-drawing devices dispelled any notions that
mathematical manipulatives were a modern invention. Van Schooten had
beaten us all to the punch more than 350 years ago.

An illustration from the title page of van Schooten’s book

As I paged through van Schooten’s book, I wondered whether his ideas
could find their way into today’s mathematics classrooms. His models,
Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Introduction • v

consisting of hinges and slotted rulers, were not always simple to build.
How could students replicate them?
Sketchpad provided a large part of the answer.
While nothing can substitute for the experience of operating a physical
model, a Sketchpad simulation comes close. In fact, Sketchpad betters
van Schooten in some ways, allowing students to manipulate the
parameters of any model and view their effects on a curve immediately.
Not every activity in this book originates from a van Schooten device, but
all share the belief that the method for producing a curve is as important
as the curve itself.
Customizing the Activities
The activities in this collection are designed with flexible learning options
in mind. Each chapter opens with an overview of the various pathways
you might follow through the material.
If constructing physical devices isn’t your interest, then skip to the
corresponding Sketchpad models. These can be built from scratch or
opened pre-made as described on page iv. If you’d like to understand
why the various curve-drawing devices work, you can follow along with
hints or strike out on your own to create an original proof. You can
complete the entire section of ellipse activities or choose a set of
interrelated activities spanning all three conics.
Any of these choices could be a sensible option for your needs.
Conic sections form the centerpiece of this book but are not the only
mathematics under study. Issues of congruency, similarity, the
Pythagorean theorem, trigonometry, and geometric optimization all
arise naturally while studying the ellipse, parabola, and hyperbola.
This book represents a marriage of 17th-century manipulatives with
21st-century technology. Enjoy the partnership.
Acknowledgments
As the preceding introduction makes clear, this book would not exist
without David Dennis. His enthusiasm and knowledge of the field
contributed immeasurably to this book’s content.
Al Cuoco and E. Paul Goldenberg from Education Development
Center, Inc. had the clever idea to use ellipses to solve optimization
problems geometrically. The optimization material in this book
draws from their chapter, “Dynamic Geometry as a Bridge from
Euclidean Geometry to Analysis,” in Geometry Turned On
(Mathematical Association of America, 1997).
Nick Jackiw suggested several of the sketches in this book. He also built
the nifty parabola-drawing device shown on the back cover.
vi • Introduction

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Bill Finzer served as my editor for the first edition of this book. His expert
guidance laid the groundwork for this revised and expanded edition. As
my new editor, Steven Chanan found ways to push the conics material in
new directions. It’s a much better book for his efforts.
Despina Stylianou and Beth Porter provided valuable feedback on the
initial draft of these materials.
Finally, I thank my New York University graduate students for their help
in testing these activities.

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Introduction • vii

Common Commands and Shortcuts
Below are some common Sketchpad actions used throughout this book. In time, these
operations will become familiar, but at first you may want to keep this list by your side.
To create a new sketch
Choose New Sketch from the File menu.
To close a sketch
Choose Close from the File menu. Or click in the
close box in the upper-left (Mac) or upper-right
(Windows) corner of the sketch.
To undo or redo a recent action
Choose Undo from the Edit menu. You can undo
as many steps as you want, all the way back to
the state your sketch was in when last opened.
To redo, choose Redo from the Edit menu.
To deselect everything
Click in any blank area of your sketch with the
Arrow tool or press Esc until objects deselect.
Do this before making selections required for a
command so that no extra objects are included.
To deselect a single object while keeping all other
objects selected, click on it with the Arrow tool.
To show or hide a label
Position the finger of the Text tool over the object
and click. The hand will turn black when it’s
correctly positioned to show or hide a label.
To change a label
Position the finger of the Text tool over the label
and double-click. The letter “A” will appear in
the hand when it’s correctly positioned.
To change an object’s line width or color
Select the object and choose from the appropriate
submenu in the Display menu.

To hide an object
Select the object and choose Hide from the
Display menu.
To construct a segment’s midpoint
Select the segment and choose Midpoint from
the Construct menu.
To construct a parallel line
Select a straight object for the new line to be
parallel to and a point for it to pass through. Then
choose Parallel Line from the Construct menu.
To construct a perpendicular line
Select a straight object for the new line to
be perpendicular to and a point for it to pass
through. Then choose Perpendicular Line from
the Construct menu.
To reflect a point (or other object)
Double-click the mirror (any straight object) or
select it and choose Mark Mirror. Then select the
point (or other object) and choose Reflect from the
Transform menu.
To trace an object
Select the object and choose Trace from the Display
menu. Do the same thing to toggle tracing off.
(If you’d rather that traces fade, check Fade Traces
Over Time on the Preferences Color panel.)
To use the calculator
Choose Calculate from the Number menu. To enter
a measurement into a calculation, click on the
measurement itself in the sketch.

Keyboard shortcuts
Command

Mac

Undo

Windows

Command

Mac

Windows

Ô+Z Ctrl+Z

Animate/Pause

Ô+`

Alt+`

Redo

Ô+R Ctrl+R

Increase Speed

Ô+]

Alt+]

Select All

Ô+A Ctrl+A

Decrease Speed

Ô+[

Alt+[

Properties

Ô+?

Midpoint

Ô+M

Ctrl+M

Hide Objects

Ô+H Ctrl+H

Intersection

Ô+I

Ctrl+I

Segment

Ô+L

Ctrl+L
Ctrl+P

Alt+?

Show/ Hide Labels Ô+K Ctrl+K
Trace Objects

Ô+T Ctrl+T

Polygon Interior Ô+P

Erase Traces

Ô+B Ctrl+B

Calculate

viii • Common Commands and Shortcuts

Ô+=

Alt+=

Action

Mac

Windows

scroll drag

Option+
Alt+drag
drag

display Context
menu

Control+
click
right-click

navigate Toolbox

Shift+arrow keys

choose Arrow,
deselect objects,
stop animations,
erase traces

Esc (escape key)

move selected
objects 1 pixel

(hold down to move
continuously)

, ⇥, ⇤, ⌅ keys

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

1

Ellipses

CHAPTER OVERVIEW
Of all the conic sections, the ellipse
is the one we see most often. Any
circle, when viewed at an angle,
appears elliptical. Tilt a drinking
glass and the water along the
surface outlines an ellipse. Place
a ball on the floor and shine
an angled beam of light onto it
from above—the ball will cast an
elliptical shadow on the floor. Slice
a cone at an angle from side to side
(as shown at right) and the cross
section is an ellipse. (This explains
why the ellipse qualifies as a conic
section.) On a grander scale, the
orbits of planets trace ellipses.
Comets in permanent orbit around
the sun follow elliptical paths.

When a plane cuts only one of two cones
forming a double-cone, the result is a
circle or an ellipse. If the cut is perpendicular to the double-cone’s axis, the result is
a circle; otherwise, it’s an ellipse.

The material in this chapter falls into three general categories:
•

Activities I–IV introduce basic ellipse terminology (focal points,
major/minor axes, eccentricity) and common ellipse construction
techniques (on paper and with Sketchpad).

•

Activities V and VI offer two lesser-known constructions that call upon
the ellipse’s distance definition for their proofs.

•

The constructions in activities VII and VIII highlight algebraic proofs.

In total, this chapter offers thirteen ellipse constructions for you to
sample. Open the multi-page sketch Ellipse Tour.gsp for a handy
slide-show overview.
I. Getting Started: When Is a Circle Not a Circle?
Introduce yourself to ellipses as a Sketchpad circle splits its center into
two. Start your ellipse explorations here.
II. The Pins-and-String Construction
Swing a string (literally!) to draw an ellipse. Then use a Sketchpad model
to formulate an ellipse’s distance definition.

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Chapter 1: Ellipses • 3

III. The Concentric Circles Construction
Two sets of concentric circles set the stage for this classic Sketchpad
ellipse construction.
IV. Some Ellipse Relationships
Investigate a pre-built Sketchpad model to uncover the algebraic and
geometric relationships between an ellipse’s major axis, minor axis,
focal length, and eccentricity.
V. The Folded Circle Construction
Watch an ellipse appear before your eyes as you fold and unfold
a paper circle. Model the technique with Sketchpad to reveal the
underlying mathematics.
VI. The Congruent Triangles Construction
A pair of congruent triangles holds the key to this unusual Sketchpad
ellipse construction.
VII. The Carpenterʼs Construction
Roll up your sleeves as you build a carpenter’s ellipsograph with a
ruler and then with Sketchpad.
VIII. Dannyʼs Ellipse
High school student Danny Vizcaino knew there had to be a better
way to construct a Sketchpad ellipse. He was right. Read all about it!
IX. Ellipse Projects
Round out the chapter with some enticing ellipse excursions.

4 • Chapter 1: Ellipses

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Getting Started: When Is a Circle Not a Circle?
Below is a circle with center at point C.

C

Imagine that the circle’s center splits into two points, F1 and F2. As the
points move away from each other, the circle deforms to accommodate
these two “centers.” The circle is now an ellipse.

F1 F2

F1

F2

F1

F2

F1

F2

F1 and F2 have special names. They’re called the focal points or foci
(singular: focus) of the ellipse.
Open the sketch Stretch.gsp in the Ellipse folder. You’ll see a circle . . .
or is it? Drag the circle’s center and watch what happens.
We draw circles by using a compass. But how can we draw ellipses?
The pins-and-string activity that follows offers one possibility.

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Chapter 1: Ellipses • 5

The Pins-and-String Construction

Name(s):

There are many ways to construct an ellipse, but perhaps the most
well-known method involves just two thumbtacks and a length of
string. You’ll use these materials now to build ellipses by hand. In
the next activity, you’ll model a similar technique with Sketchpad.
Constructing a Physical Model
Preparation: You’ll need a piece of string,
two thumbtacks, a corkboard, a large sheet
of paper, and a pencil.
1. Place your paper on top of the
corkboard and stick the two
thumbtacks into it. Then tie each
end of the string onto a thumbtack.
Pick any locations for the thumbtacks,
but make sure to leave some slack in
the string. The thumbtacks will be the
focal points of your ellipse.

From Sive de Organica Conicarum
Sectionum in Plano Descriptione,
Tractatus, by Dutch mathematician
Frans van Schooten, 1646.

2. Pull the string taut with your pencil.
Make sure the string lies near the tip of the pencil.
3. Keeping the string taut, swing the pencil around the focal points,
letting the tip of the pencil trace its path. You may need to reposition
the pencil to draw the entire curve.
Questions
Q1

What symmetries do you see in your ellipse? Draw a picture to
illustrate any lines of symmetry.

Q2

Move the ends of your string so that they’re farther apart.
Redraw the ellipse and describe how its shape compares to your
original curve.

Q3

Move the ends of your string close together. Redraw the ellipse and
describe how its shape compares to your original curve.

Q4

Suppose you move the ends of your string so far apart that the string
is fully extended and taut. What “curve” will your pencil draw?

Q5

Suppose you attach both ends of the string to the same thumbtack.
What curve will your pencil draw?

Q6

Suppose a friend wants to draw an ellipse identical to one of yours.
What two pieces of information would you need to give her so that
she could reproduce it?

6 • Chapter 1: Ellipses

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

The Pins-and-String Construction (continued)
Q7

There are two important segments associated with every ellipse: the
major axis and the minor axis. In each of the following illustrations,
segment AB is the major axis, segment CD is the minor axis, and F1
and F2 are the focal points.
D

B
F2

A

F1

F2

B

C

D

D

C

B
F2

A

F1

F1
C

A

Based on these pictures, write your own definition of the major and
minor axes.
Q8

Without using any tools, how could you form the major and minor
axes of an ellipse cut out of paper?

Uncover the Imposter
Open the sketch Points.gsp in the Ellipse folder. You’ll see two foci, F1
and F2, of an invisible ellipse and three points, A, B, and C. The ellipse
passes through exactly two of these three points, but which ones?
Use Sketchpad’s measurement and calculation tools to spot the imposter.
Q9
Q10

Which point doesn’t sit on the ellipse? Explain how you can tell.
Complete this sentence without mentioning of pins or string:
An ellipse is the set of points P such that the following value is
constant for all locations of P:

Explore More
1. Open the sketch Parametric Ellipses.gsp (Ellipse folder) to view
ellipses constructed with Sketchpad’s parametric coloring feature.
2. Here is the definition of a new curve similar to an ellipse:
The set of points P such that PA + PB + PC is constant for
three fixed points, A, B, and C.

Open the sketch New Curve.gsp (Ellipse folder) to see how
such a curve can be constructed using Sketchpad’s parametric
coloring feature.
Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Chapter 1: Ellipses • 7

The Concentric Circles Construction

Name(s):

Circles that are concentric share the same center. In this activity, you’ll
use two sets of concentric circles to draw an ellipse by hand. You’ll then
transfer this technique to Sketchpad to draw an ellipse whose shape and
size can be adjusted just by dragging your mouse.
Sketching Ellipses by Hand
The illustration below shows two sets of concentric circles. One set of
circles is centered at point F1, the other at point F2. For each set, the radii
of the circles increase by 1’s, from 1 unit all the way up to 6 units.
Points F1 and F2 are the foci of an infinite number of ellipses, but only two
that we’re interested in: the one that passes through point A and the one
that passes through point B.

Remember: An
ellipse is the set
of points P such
that PF1 + PF2
is constant.

Q1

How many units apart are points A and F1? How many units apart are
points A and F2? What is the numerical value of AF1 + AF2?

Q2

Locate and mark at least seven points that sit on the ellipse passing
through point A. Explain how you found them.

Q3

Locate and mark at least seven other points that sit on the ellipse
passing through point B. (Use a different colored pen or pencil if
possible.) Explain how you found them.

Q4

Using the points you found as guidelines, sketch the two ellipses.

B

A
F1

8 • Chapter 1: Ellipses

F2

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

The Concentric Circles Construction (continued)
Constructing a Sketchpad Model
Now that you’ve drawn some ellipses by hand, it’s time to build a
dynamic one: an ellipse that changes shape as its parts are dragged.
Follow the steps below to construct a Sketchpad ellipse. As you do, think
about how this method relates to the concentric circles technique.
A

F1

C

B

F2

1. Draw a horizontal line near the top of your screen. Hide the
control points by selecting them and choosing Hide Points
from the Display menu.
2. Construct points A, B, and C on the line. The location of these points
doesn’t matter, but make sure point C is between points A and B.
3. Hide the line. Draw individual segments AC and CB.
4. Draw points F1 and F2 to represent the foci of your ellipse.
5. Using the Circle by Center+Radius command, construct a circle with
center F1 and radius AC. Construct another circle with center F2 and
radius CB.
6. Construct the intersection points of the circles. (You might have to
adjust your model so the circles intersect.)
7. Select the intersection points and choose Trace Intersections from the
Display menu.
If you don’t want
your traces to fade,
be sure the Fade
Traces Over Time
box is unchecked
on the Color panel
of the Preferences
dialog box.

8. Drag point C back and forth slowly along AB and observe the trace of
the two points.
9. Change the distance between F1 and F2. Then, if necessary, choose
Erase Traces from the Display menu to erase your previous curve.
Trace several new curves, each time varying the distance between the
focal points.

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Chapter 1: Ellipses • 9

The Concentric Circles Construction (continued)
For every new location of F1 and F2, you need to retrace your curve.
Ideally, your ellipse should adjust automatically as you drag either focus.
Sketchpad’s powerful Locus command makes this possible.
10. Turn tracing off for the two intersection points by selecting them and
once again choosing Trace Intersections from the Display menu.
11. Select one of the two intersection points and point C. Choose Locus
from the Construct menu. Do this again for the other intersection
point and point C. You’ll form an entire curve: the locus of the
two intersection points as point C moves along segment AB.
Drag F1, F2, or point B to change the size and shape of the ellipse.
Questions
Q5

As you drag point C along segment AB, the radii of both circles
change lengths. Still, there is a relationship that exists between the
two radii regardless of point C’s position. What is it?

Q6

Explain why the two intersection points of the two circles trace
an ellipse.

Q7

How far apart can the two focal points be before you can no longer
trace an ellipse?

Q8

Select your two onscreen circles, choose Trace Circles from the
Display menu, then drag point C along segment AB.
Based on this experiment, describe the similarities between your
Sketchpad construction and the concentric circles technique.

Explore More
1. By shortening the distance between points A and B and dragging
point C to the left of A and to the right of B so that it does not lie
between them, it’s possible to draw a different type of curve. Try it
and see what you get.
2. Consider the definition of a constant-perimeter rectangle:
A constant-perimeter rectangle is a rectangle constructed in
Sketchpad whose dimensions can change, but whose
perimeter always remains fixed at a given, constant value.

This means that a rectangle with constant perimeter of 20 inches
could have dimensions of 3”⇥ 7”, 6”⇥ 4”, or 2”⇥ 8”, but not 5”⇥ 6”.
Use the techniques you learned when building a Sketchpad ellipse
to construct a constant-perimeter rectangle.

10 • Chapter 1: Ellipses

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Some Ellipse Relationships

Name(s):

In the Pins-and-String activity, you drew ellipses with a pencil and a
taut piece of string. Now, you’ll use Sketchpad to explore some of the
algebraic and geometric relationships that exist between this string and
an ellipse’s major and minor axes.
Finding Lengths
Open the sketch String.gsp in the Ellipse folder. You’ll see an ellipse with
major axis AB, minor axis CD, and focal points at F1 and F2.
Point P represents the pencil point that’s pulling taut a string attached to
points F1 and F2. Together, segments PF1 and PF2 represent the string.
The total length of the “string” is 20 units (PF1 + PF2). The distance
between F1 and F2 is 16 units.
D
P

A

F1

F2

B

C

Questions
Answer these questions without taking any measurements with Sketchpad.
For each one, drag point P around the ellipse until you find a location that
helps you answer the question.
Q1

What is the length of the major axis AB? Explain where you
positioned point P to reach your conclusion.

Q2

What is the length of the minor axis CD? Explain where you
positioned point P to reach your conclusion.

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Chapter 1: Ellipses • 11

Some Ellipse Relationships (continued)
Eccentricity
Some ellipses are “skinny” and elongated. Others are “fat” and nearly
circular. The eccentricity of an ellipse is a measure that captures the shape
of an ellipse in one numerical value.
The eccentricity of an ellipse is defined as:
the distance between the focal points
the distance between the endpoints of the major axis
1. Open the sketch Eccentricity.gsp in the Ellipse folder. You’ll see an
ellipse with major axis endpoints A and B, and focal points F1 and F2.
2. Use Sketchpad’s Distance command to measure the two distances
needed to compute the ellipse’s eccentricity.
3. Use Sketchpad’s Calculate command to compute the eccentricity.
Questions
Q3

Your Sketchpad ellipse can change its size and shape to represent a
whole collection of ellipses. Before experimenting with it, make a
prediction: How small and how large do you think an ellipse’s
eccentricity can become? Explain your reasoning.

Q4

Drag point B to change the size and shape of the ellipse. As you do so,
monitor the values of its eccentricity. Does your prediction from Q3
hold? Modify it if necessary.

Q5

How many ellipses can share the same eccentricity? How could
you create two ellipses with the same eccentricity without using
Sketchpad?

Explore More
See Q2 from earlier
in the activity
for help with
this construction.

1. Open the second page of the sketch String.gsp. You’ll see a new
ellipse along with the length of “string” used to draw it. Construct
the focal points of the ellipse. No measuring allowed!
Make sure your foci are dynamic: they should adjust themselves to
remain in the proper locations as you change the length of the string.
2. Johannes Kepler’s First Law of planetary motion states that the orbit
of each planet is an ellipse with the Sun at one focus. Do some
research to find the eccentricities of our planets. Why might
astronomers before Kepler have believed planets moved in
circular motion?

12 • Chapter 1: Ellipses

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

The Folded Circle Construction

Name(s):

Sometimes a conic section appears in the unlikeliest of places. In this
activity, you’ll explore a paper-folding construction in which crease lines
interact in a surprising way to form a conic.
Constructing a Physical Model
Preparation: Use a compass to draw a circle with a radius of approximately
three inches on a piece of wax paper or patty paper. Cut out the circle with
a pair of scissors. (If you don’t have these materials, you can draw the circle
in Sketchpad and print it.)
1. Mark point A, the center of your circle.
If you’re working
in a class, have
members place B at
different distances
from the center.
If you’re working
alone, do this
section twice—
once with B close
to the center,
once with B close
to the edge.

2. Mark a random point B within the
interior of your circle.
3. As shown below right, fold
the circle so that a point on its
circumference lands directly onto
point B. Make a sharp crease to
keep a record of this fold.
Unfold the circle.

B
A

4. Fold the circle along a new crease
so that a different point on the
circumference lands on point B. Unfold
the circle and repeat the process.
5. After you’ve made a dozen or so creases,
examine them to see if you spot any
emerging patterns.
Mathematicians
would describe
your set of creases
as an envelope
of creases.

B
A

6. Resume creasing your circle. Gradually,
a well-outlined curve will appear. Be patient—
it may take a little while.
7. Discuss what you see with your
classmates and compare their folded
curves to yours. If you’re doing this
activity alone, fold a second circle with
point B in a different location.

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Chapter 1: Ellipses • 13

The Folded Circle Construction (continued)
Questions
Q1

The creases on your circle seem to form the outline of an ellipse.
What appear to be its focal points?

Q2

If you were to move point B closer to the edge of the circle and fold
another curve, how do you think its shape would compare to the
first curve?

Q3

If you were to move point B closer to the center of the circle and fold
another curve, how do you think its shape would compare to the
first curve?

Constructing a Sketchpad Model
Fold and unfold. Fold and unfold. Creasing your circle takes some work.
Folding one or two sheets is fun, but what would happen if you wanted
to continue testing different locations for point B? You’d need to keep
starting with fresh circles, folding new sets of creases.
Sketchpad can streamline your work. With just one circle and one set of
creases, you can drag point B to new locations and watch the crease lines
adjust themselves instantaneously.
C

8. Open a new sketch and use the
Compass tool to draw a large
circle with center A. Hide the
circle’s radius point.

B

9. Use the Point tool to draw
a point B at a random spot
inside the circle.

A
crease

10. Construct a point C on the
circle’s circumference.
11. Construct the “crease” formed
when point C is folded onto point B.
12. Drag point C around the circle. If you constructed your crease line
correctly, it should adjust to the new locations of point C.
13. Select the crease line and choose Trace Line from the Display menu.
If you don’t want
your traces to fade,
be sure the Fade
Traces Over Time
box is unchecked
on the Color panel
of the Preferences
dialog box.

14. Drag point C around the circle to create a collection of crease lines.
15. Drag point B to a different location and then, if necessary, choose
Erase Traces from the Display menu.
16. Drag point C around the circle to create another collection of
crease lines.

14 • Chapter 1: Ellipses

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

The Folded Circle Construction (continued)
Retracing creases for each location of point B is certainly faster than
folding new circles. But we can do better. Ideally, your crease lines should
relocate automatically as you drag point B. Sketchpad’s powerful Locus
command makes this possible.
17. Turn tracing off for your original crease line by selecting it and once
again choosing Trace Line from the Display menu.
18. Now select your crease line and point C. Choose Locus from the
Construct menu. An entire set of creases will appear: the locus of
crease locations as point C moves along its path. If you drag point B,
you’ll see that the crease lines readjust automatically.
19. Save your sketch as Creased Circle.gsp. You’ll use it again in one of
the Hyperbola Projects activities.
Questions

The Merge and
Split commands
appear in the
Edit menu.

Q4

How does the shape of the curve change as you move point B closer to
the edge of the circle?

Q5

How does the shape of the curve change as you move point B closer to
the center of the circle?

Q6

Select point B and the circle. Then merge point B onto the circle’s
circumference. Describe the crease pattern.

Q7

Select point B and split it from the circle’s circumference. Then merge
it with the circle’s center. Describe the crease pattern.

Playing Detective
Each crease line on your circle touches the ellipse at exactly one point.
Another way of saying this is that each crease is tangent to the ellipse. By
engaging in some detective work, you can locate these tangency points
and use them to construct just the ellipse without its creases.
20. Open the sketch Folded Circle.gsp in the Ellipse folder. You’ll see a
thick crease line and its locus already in place.
21. Drag point C and notice that the crease line remains tangent to the
ellipse. The exact point of tangency lies at the intersection of two
lines—the crease line and another line not shown here. Construct
this line in your sketch as well as the point of tangency, point E.
Select the locus
and make its width
thicker so that it’s
easier to see.

22. Select point E and point C and choose Locus from the Construct
menu. If you’ve identified the tangency point correctly, you
should see a curve appear precisely in the white space bordered
by the creases.

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Chapter 1: Ellipses • 15

The Folded Circle Construction (continued)
How to Prove It
C

The Folded Circle construction
seems to generate ellipses. Can
you prove that it does? Try
developing a proof on your
own, or work through the
following steps and questions.
The picture at right should
resemble your construction.
Line HI (the perpendicular
bisector of segment CB)
represents the crease formed
when point C is folded onto
point B. Point E sits on the
curve itself.

I

E
B
A

H

23. Add segments CB, BE, and AC to the picture.
24. Label the intersection of CB with the crease line as point D.
Questions

Remember: An
ellipse is the set
of points such
that the sum of the
distances from each
point to two fixed
points (the foci)
is constant.

Q8

Use a triangle congruence theorem to prove that jBED m jCED.

Q9

Segment BE is equal in length to which other segment? Why?

Q10

Use the distance definition of an ellipse and the result from Q9 to
prove that point E traces an ellipse.

Explore More
1. When point B lies within its circle, the creases outline an ellipse.
What happens when point B lies outside its circle?
2. Use the illustration from your ellipse proof to show that
⇧AEH = ⇧BED.
Here’s an interesting consequence of this result: Imagine a pool
table in the shape of an ellipse with a hole at one of its focal points.
If you place a ball on the other focal point and hit it in any direction
without spin, the ball will bounce off the side and go straight into
the hole. Guaranteed!

16 • Chapter 1: Ellipses

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

The Congruent Triangles Construction
A linkage is any
device with hinged
and slotted rods.

Name(s):

The picture below appears in a book by the seventeenth-century
mathematician Frans van Schooten. It shows a linkage consisting of
three movable rods hinged together. In this activity, you’ll explore
a Sketchpad model of van Schooten’s device and prove that it
draws ellipses.
1. Open the sketch Congruent
Triangles.gsp in the
Ellipse folder. You’ll
see a construction that
matches most of van
Schooten’s picture.
This sketch was created so that
two equalities always hold.
These are:
AB = FC
BF = CA

C

F

G

E

A

B

2. Drag point F. As you do,
observe how the color-coded
segments remain fixed in size
and equal to each other.
3. Select point E and choose Trace Intersection from the Display menu.
Drag point F and observe the curve traced by point E.
4. You’ve drawn what looks like half an ellipse. To trace the other
half, use Sketchpad’s Reflect command to reflect point E across
segment AB. Select the reflected point, E , and choose Trace Point.
Now drag point F to trace the entire curve.
Questions
Q1

What appear to be the focal points of your ellipse?

Q2

The blue segment at the bottom left of your screen controls the lengths
of segments AB and FC. Drag its right endpoint to lengthen or shorten
it. Choose Erase Traces from the Display menu to remove your
previous curve. Now, retrace your curve. Do this several times.
How does the length of the blue segment affect your curve?

Q3

If you look at van Schooten’s picture, you’ll see that it includes a
point G. Construct lines through segments AB and FC on your sketch
so they meet at point G. Now draw a line through points E and G.
Drag point F.
What is the relationship between this newly created line and
your curve?

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Chapter 1: Ellipses • 17

The Congruent Triangles Construction (continued)
How to Prove It
It certainly looks like the Congruent Triangles construction draws ellipses,
but can you explain why? Try developing a proof on your own, or work
through the following questions.
Questions
Q4

Add segment AF to your sketch. Using a triangle congruence theorem,
show that jABF m jFCA.

Q5

Use the result from the question above to complete this statement:
⇧FCA = ⇧

Remember: An
ellipse is the set of
points such that the
sum of the distances
from each point to
two fixed points (the
foci) is constant.

Q6

Use the angle equality above and other information you know about
the linkage to show that jAEB m jFEC.

Q7

Segment AE is equal in length to which other segment? Why?

Q8

Use the distance definition of an ellipse and the result from Q7 to
prove that point E traces an ellipse.

Explore More
1. This construction is sometimes called the crossed parallelogram.
Explain why.
2. Use Sketchpad’s Trace command to display the locus of point C as
you drag it. Describe all of the similarities you can find between this
sketch and the Folded Circle construction.
You’ll need to
make liberal use
of Sketchpad’s
Circle by
Center+Radius
command.

3. Open the sketch Gears.gsp in the Ellipse folder to operate a pair of
elliptic gears.
4. Starting with a blank sketch, build your own Sketchpad model of the
Congruent Triangles construction.

18 • Chapter 1: Ellipses

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

The Carpenter’s Construction

Name(s):

The device pictured at right, a
favorite among carpenters and
woodworkers, is called an
ellipsograph or trammel.
The ellipsograph
first appears in the
work of Proclus
(A.D. 410–485).

An ellipsograph has an arm
with two bolts that slide along
a pair of perpendicular tracks.
As the bolts glide along their
respective grooves, a pen attached
to one end draws an ellipse.
In this activity, you’ll build your own ellipsograph with a ruler, then
explore a more robust model with Sketchpad. Finally, you’ll prove that
this device really does draw ellipses.
Constructing a Physical Model
Preparation: You’ll need a ruler, some masking tape, a pen or pencil, and a large
piece of paper (11”⇥ 17” is good, but 8.5” ⇥ 11” works also).

If you’re working
in a class, have
members pick different
placements of B
relative to A and C.
If you’re working
alone, do this activity
twice with different
point B locations.

1. Put a long strip of masking tape on a
ruler, lining up an edge of the tape with
an edge of the ruler. Mark three points,
A, B, and C, on the tape. Make sure the
distance between points A and C is less
than half the width of your paper.

A

B

C

2. Use your ruler to draw a pair of
perpendicular lines on the paper.
The illustration on the next page shows
how to slide your ruler along the lines.
3. Begin by positioning point B at the
intersection of the two lines and
point A on the horizontal line to
the left of B. Place a mark on your
paper at point C (also on the
horizontal line for now).
4. Slide the ruler just a little so that
point A continues to lie on the
horizontal line and point B lies on
the vertical line. Mark the location of point C.
5. Continue to slide points A and B in small increments, keeping
point A on the horizontal line and point B on the vertical line. Each
time you reposition the ruler, mark point C’s position. Eventually the
ruler will sit vertically with point A at the intersection point.
Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Chapter 1: Ellipses • 19

The Carpenterʼs Construction (continued)
6. You’ve drawn what appears to be a quarter ellipse. Continue
repositioning the ruler to draw the entire curve.
7. Discuss what you see with your classmates and compare their curves
to yours. If you’re doing this activity alone, draw a new curve by
changing the location of point B.

C
A

B

C

A

B

C
B
A

Questions
Q1

Shown below are two rulers with different relative locations for
points A, B, and C. If each ruler is used to draw an ellipse, how will
the shape of the two curves differ?
A

A

B

C

B

C

Q2

Assuming that ellipsographs do draw ellipses, how would you
position points A, B, and C to draw an ellipse with a 20-cm major
axis and a 12-cm minor axis?

Q3

Can an ellipsograph draw circles? Explain why or why not.

20 • Chapter 1: Ellipses

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

The Carpenterʼs Construction (continued)
Investigating a Sketchpad Model
Your ruler model of an ellipsograph provides a good sense of how the
device works. Yet sliding the ruler bit by bit was probably awkward.
A Sketchpad ellipsograph offers some advantages: a smooth, continuous
motion and easily adjustable lengths.
8. Open the sketch Carpenter.gsp
in the Ellipse folder.
C

9. Drag point A. As you do, watch
the motion of segment AC.
B

10. Select point C and choose
Trace Intersection from the
Display menu. Drag point A
and observe the curve traced
by point C.
The curve you see is the locus
of point C as point A moves
along its line.

A

11. You’ve drawn what looks like half an ellipse. To trace the other half,
use Sketchpad’s Reflect command to reflect point C across the
horizontal axis. Select the reflected point, C , and choose Trace Point.
Now drag point A to trace the entire curve.
12. Adjust the lengths of AB and BC at the bottom left of your screen to
vary the parameters of the ellipsograph. Then choose Erase Traces
from the Display menu to erase your previous curve. Trace several
new curves, each time varying the parameters AB and BC.
13. Turn tracing off for points C and C by selecting them and once again
choosing Trace Points from the Display menu.
14. Now select points A and C. Choose Locus from the Construct menu.
Do this again for points A and C . You’ll form an entire curve: the
locus of points C and C .
15. As before, adjust the lengths of AB and BC to observe their effect on
the curve.

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Chapter 1: Ellipses • 21

The Carpenterʼs Construction (continued)
How to Prove It
The curves you’ve drawn certainly look like ellipses. But appearances
don’t always tell the whole story. A proof can confirm the curves’
identities and provide insights into the mathematics underlying
ellipsographs.
Do you know the algebraic representation of an ellipse? Take a look at the
following definition:
The points satisfying the equation
x2
a2

+

y2
b2

= 1, with b > a,

lie on an ellipse centered at the origin with major axis of length 2b
along the y-axis and minor axis of length 2a along the x-axis.
The illustration below shows an ellipsograph in the x-y plane whose arm
is represented by segment AC. For this particular ellipsograph, AB = 6 and
BC = 3. Several extra segments are included in the picture: segment CE is
parallel to the y-axis and segment BD is parallel to the x-axis.
Since the location of point C changes as the ellipsograph’s arm moves, it’s
labeled as (x, y), using variables as coordinates. If C traces an ellipse, you
should be able to derive an equation like the one above relating x to y.
Questions
The questions that follow provide a step-by-step guided proof. You can
answer them or write your own proof without any hints.

C = (x, y)
3

B

D

6

A

22 • Chapter 1: Ellipses

(0, 0)

E

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

The Carpenterʼs Construction (continued)
Hint: Determine
the lengths of the
ellipse’s major and
minor axes, and the
location of its center.

Q4

Assume for a moment—without proof—that the ellipsograph in
the picture on the previous page draws an ellipse. Given that AB = 6
and BC = 3, what is its equation? (We’ll compare our eventual answer
with this equation later.)

Q5

Fill in the lengths of the following segments in terms of x and y:
BD =
CE =
CD =
Explain how you found the length of CD.

Q6

Now that you’ve determined the lengths of various segments, find a
way to relate x to y. Here are two approaches you might consider:
• Look for a pair of similar triangles in the diagram. Use their
similarity to create a proportion relating x to y.
• Compute sin(⇧CAE) and cos(⇧CBD). Look for a way to relate
these two values to each other.
If necessary, manipulate your equation so that it’s recognizable as that
of an ellipse. Compare your equation to the one you found in Q4 to
see if they match.

Q7

Rewrite your proof, this time making it more general. Let AB = s
and BC = t.

Explore More
1. Starting with a blank sketch, build your own Sketchpad model of the
ellipsograph construction.
2. Given any curve drawn by an ellipsograph, you should be able to find
its foci. Open page 3 of the sketch Carpenter.gsp. Use what you’ve
learned in the Some Ellipse Relationships activity to construct the foci
of the ellipse on screen.
Make sure your foci are dynamic: they should adjust to remain in the
proper locations as you change the lengths of segments AB and BC.

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Chapter 1: Ellipses • 23

The Carpenterʼs Construction (continued)
For added drama,
imagine that you’re
standing on the
ladder. What path
does your foot trace?

3. The illustrations below show a seventeenth-century drawing device
whose motion resembles that of a ladder sliding down a wall. On
the ladder sits a bucket. As the ladder slides, what path does the
bucket trace?

Investigate this problem by modifying your physical model of
the ellipsograph. Build a Sketchpad model, too. Can you prove
your findings?
4. There’s a theorem from geometry that states:
The midpoint of the hypotenuse of a right triangle is
equidistant from the three vertices of the triangle.

Assume for the moment that this statement is true. How can you use
it (and nothing else) to prove that the bucket from Q3 traces a circle
when it’s midway up the ladder?
5. Leonardo da Vinci devised an
ellipse-tracing technique that
substitutes a sliding triangle for
the sliding ellipsograph.

C
B

Open the sketch Triangle.gsp in
the Ellipse folder. Examine the
path traced by triangle vertex C
as the other two vertices slide
along the x- and y-axes.

24 • Chapter 1: Ellipses

A

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Danny’s Ellipse

Name(s):

While a student at Mountain View High School in California,
Danny Vizcaino devised a novel way to construct ellipses with
Sketchpad. In this activity, you’ll build a model of Danny’s ellipse,
then prove why his method works.
Constructing a Sketchpad Model
1. Open a new sketch.
Choose Show Grid from
the Graph menu. Then
choose Hide Grid to
remove the grid lines while
keeping the x- and y-axes.

D
B
E
F

2. Label the origin (the
intersection of the two
lines) as point A.
3. Draw a random point B
on the y-axis and a
point C on the x-axis.

A

C

c1
c2

4. Select, in order, points A
and B. Then choose
Circle by Center+Point from the Construct menu to build a
circle c1 with center at point A passing through point B.
5. Repeat step 4 to construct a circle c2 with center at point A passing
through point C.
6. Draw a segment from point A to a random point D on circle c2.
7. Construct point E, the intersection of segment AD with circle c1.
8. Construct a line through point D perpendicular to the x-axis.
9. Construct a line through point E perpendicular to the y-axis.
10. Construct point F, the intersection of the two lines you just created.
If you don’t want
your traces to fade,
be sure the Fade
Traces Over Time
box is unchecked
on the Color panel
of the Preferences
dialog box.

11. Select point F and choose Trace Intersection from the Display menu.
Drag point D around its circle and observe the curve traced by
point F.
The curve you see is the locus of point F as point D moves around
its circle.

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Chapter 1: Ellipses • 25

Dannyʼs Ellipse (continued)
12. Drag points B and C to alter the sizes of circles c1 and c2. Then, if
necessary, choose Erase Traces from the Display menu to erase your
previous curve. Trace several new curves, each time varying the sizes
of the two circles.
For every new location of points B and C, you need to retrace your curve.
Ideally, your ellipse should adjust itself automatically. Sketchpad’s
powerful Locus command makes this possible.
13. Turn tracing off for point F by selecting it and once again choosing
Trace Intersection from the Display menu.
14. Now select points D and F. Choose Locus from the Construct menu.
You’ll form an entire curve: the locus of point F. Drag points B and C
to vary the shape of the curve.
How to Prove It
Open Dynamic
Geometry.gsp in
the Ellipse folder to
see a fun application
of Danny’s method.

Intrigued by Danny’s Sketchpad construction, Key Curriculum Press
sponsored a worldwide contest to answer the following question:
Danny’s curve looks like an ellipse, but is it?
The contest is over, but the challenge remains. Can you prove that
Danny’s curve is an ellipse? To do so, you’ll need the algebraic
definition below.
The points satisfying the equation
x2
a2

+

y2
b2

= 1, with a > b,

lie on an ellipse centered at the origin with major axis of length 2a
along the x-axis and minor axis of length 2b along the y-axis.
In the illustration on the next page, AE = 2 and ED = 3. Segment EH is
perpendicular to the x-axis. Since the location of point F changes as
point D moves, it’s labeled as (x, y), using variables as coordinates.
If F traces an ellipse, you should be able to derive an equation like
the one above relating x to y.

26 • Chapter 1: Ellipses

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Dannyʼs Ellipse (continued)
Questions
The questions that follow provide a step-by-step guided proof. You can
answer them or write your own proof without any hints.
Q1

Assume for a moment—without proof—that Danny’s curve is an
ellipse. What is its equation? (We’ll compare our eventual answer
with this equation later.)
Hint: Determine the
lengths of the ellipse’s
major and minor axes and
the location of its center.

For Q1 and Q2,
use the numerical
values in the picture:
AE = 2 and ED = 3.

Q2

D
3
E

F = (x, y)

2

Fill in the lengths of the
following segments in
terms of x and y:

A

H

G

c1

AG =
EH =

c2

DG =

Explain how you found the length of DG .
Q3

Now that you’ve determined the lengths of various segments, find a
way to relate x to y. Here are two approaches you might consider:
• Look for a pair of similar triangles in the diagram. Use their
similarity to create a proportion relating x to y.
• Compute sin(⇧EAH) and cos(⇧DAG). Look for a way to relate
these two values to each other.
If necessary, manipulate your equation so that it’s recognizable as that
of an ellipse. Compare your equation to the one you found in Q1 to
see if they match.

Q4

Rewrite your proof, this time making it more general. Let AE = s
and ED = t.

Explore More
See page 3
of Danny.gsp
for help.

1. Since Danny’s curve is an ellipse, you should be able to find its foci.
Open the second page of the sketch Danny.gsp in the Ellipse folder.
Use what you’ve learned in the Some Ellipse Relationships activity to
construct the foci of the ellipse on screen.
Make sure your foci are dynamic: they should adjust to remain in the
proper locations as you drag point B or C.

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Chapter 1: Ellipses • 27

Ellipse Projects
The projects below extend your ellipse knowledge in new directions and
are ideal for in-class presentations.
1. Recall the distance definition of an ellipse:
An ellipse is the set of point P such that PA + PB
is constant for two fixed points, A and B.

Suppose we define a new curve with a similar description:
The set of points P such that PA + 2PB is constant
for two fixed points, A and B.

What does this curve look like? Build a Sketchpad model by
modifying the Concentric Circles construction.
2. In the Folded Circle construction, you built the crease line formed
when point C is folded onto point B by constructing the perpendicular
bisector of segment BC. Imagine now that Sketchpad’s Midpoint and
Perpendicular Line commands are broken. How can you construct
the crease line without them?
The picture below offers one possibility. It’s from the seventeenthcentury mathematician Frans van Schooten and shows a rhombus
FCGB with a slotted rod passing through points F and G.
Open the sketch Rhombus.gsp in the Ellipse folder and experiment
with the model. How is this model similar to the Folded Circle
construction? What purpose does the rhombus serve?

C

G

E

F
A

28 • Chapter 1: Ellipses

B

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Ellipse Projects (continued)
3. The book Feynman’s Lost Lecture: The Motion of Planets Around the Sun
(W. W. Norton & Company, 2000) features a lecture by legendary
physicist Richard Feynman. In his talk, Feynman uses the Folded
Circle construction to demonstrate geometrically that planets orbit the
sun in elliptic paths. Read Feynman’s lecture and prepare a report on
his method.
4. Build your own working model of the Congruent Triangles
construction using cardboard and paper fasteners.
What happens
when AB ⌅ BC ?

5. Open the sketch Bent Straw.gsp in the Ellipse folder. You’ll see a
linkage with equal-length segments AB and BC. As you drag point C,
notice how the motion of this device resembles that of a bent straw.
Can you prove point D traces an ellipse? The sketch contains some
suggestions to get you started.
B
D

A
C

You’ll use this tool in
the Burning Tent
activity later
in this book.

6. Use techniques from the Some Ellipse Relationships activity as
well as from Danny’s Ellipse to build a custom tool that takes
three points—A, B, and P—and constructs an ellipse passing
through P with A and B as foci.
If you need help, the sketch Foci and Point.gsp in the Ellipse folder
provides assistance.
7. The sketch Tangent Circles.gsp in the Ellipse folder shows a red
circle c3 that’s simultaneously tangent to circles c1 and c2. Press the
Animate button and observe the path of point C, the center of circle c3.
Can you prove that C traces an ellipse?

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Chapter 1: Ellipses • 29

2
Parabolas

CHAPTER OVERVIEW
The path of a baseball. The curve
formed by the cables on the Golden
Gate Bridge. The trail of water jetting
out from a hose. All of these are
examples of parabolic curves (or very
nearly so).
The picture at right shows a cone
that’s been sliced by a plane parallel
to a side. The cross section (the “conic
section”) is a parabola.
This chapter opens with a look at
a parabola’s focus and directrix
(activity I). It then presents two
parabola constructions, one based
on the parabola’s distance definition
(activity II) and the other on a
parabola’s algebraic form (activity III).

When a double-cone is cut by a plane
that’s parallel to an edge, the result is
a parabola.

In total, this chapter offers seven parabola constructions for you to
sample. Open the multi-page sketch Parabola Tour.gsp for a handy
slide-show overview.
I. Introducing the Parabola
A focal point and a directrix line are the main ingredients for
two parabola constructions.
II. The Folded Rectangle Construction
With just a blank sheet of paper and a single point, you can fold
yourself a genuine parabola. Model the technique with Sketchpad
to reveal the underlying mathematics.
III. The Expanding Circle Construction
As a circle grows and shrinks, it defines points that lie along a parabola.
Investigate this tenth-century construction with the aid of Sketchpad.
IV. Parabola Projects
Round out the chapter with some pleasing parabola projects.

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Chapter 2: Parabolas • 33

Introducing the Parabola

Name(s):

In this activity, the geometric definition of a parabola serves as a gateway
for investigating two different ways to construct the curve.
Defining a Parabola
Below is the geometric definition of a parabola:
A parabola is the set of points equidistant from a fixed point
(the focus) and a fixed line (the directrix).
If you open the sketch Parabola.gsp in the Parabola folder, you’ll see a
parabola along with its focus and directrix. The sketch also contains two
measurements showing the distance of point P from the focus and from
the directrix.
Drag point P. You’ll see the distance measurements change, but always
remain equal to each other.
The parabola is a symmetric curve. Its line of symmetry passes through
the focus and through a point called the vertex. In the parabola below,
the vertex is the lowest point on the curve.

focus

vertex

directrix

Questions

Alternatively,
press the show
segments button.

Q1

Given a line d and a point P not on the line, how do we define the
distance between them?

Q2

Draw a segment from point P to the focus. Then construct a segment
whose length represents the distance from point P to the directrix.
The segments should adjust themselves as point P moves along
the parabola.
Use Sketchpad to measure the lengths of these two segments. What
do you expect to find?

34 • Chapter 2: Parabolas

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Introducing the Parabola (continued)
Q3

Given just a parabola’s focus and directrix, how can you construct
its vertex?

The Concentric Circles Method
Concentric circles
share the
same center.

The illustration below shows nine concentric circles centered at point A.
The radii of the circles increase by 1’s, from 1 unit all the way up to
9 units. The horizontal lines are also spaced 1 unit apart. Each line,
except the one passing through point A, is tangent to a circle.
We can use this arrangement to draw parabolas.

B

4
3

A

2
1

Questions
Q4

How many units apart are points A and B? How many units apart
are point B and line 1? Based on these measurements, what can
you conclude?

Q5

Locate and mark at least 15 points (including the vertex) that sit on a
parabola with focal point at A and line 1 as its directrix. Explain how
you found them.

Q6

Using the points you found as guidelines, sketch the parabola.

Q7

Repeat the previous two questions, drawing parabolas with directrix
lines 2, 3, and 4. All three parabolas will have A as their focal point.
For each parabola, use a different colored pen or pencil if possible.

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Chapter 2: Parabolas • 35

Introducing the Parabola (continued)
The Sliding Ruler Method
Open the sketch Sliding Ruler.gsp in the Parabola folder. You’ll see the
model below.
The rectangles represent two rulers. A piece of string equal in length to
BD is attached from point A to the corner of the vertical ruler (point B).
The string is held taut against the edge of the ruler by a pencil at point C.
Sliding ruler BD to the right while keeping the string taut causes point C
to trace half a parabola.
To operate the Sketchpad model, drag point D.
B

C

A

D

How to Prove It
The Sliding Ruler construction seems to draw parabolas. Can you prove
that it does? Try developing a proof on your own or work through the
following questions.
Questions

Remember: The
length of string is
equal to the length
of ruler BD.

Q8

Assuming the pencil at point C traces a parabola, where are the focus
and directrix?

Q9

Assuming the pencil at point C traces a parabola, which two segments
must you prove equal in length?

Q10

Complete this statement:
BC + CA = BC +

Q11

36 • Chapter 2: Parabolas

Complete the proof.

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Introducing the Parabola (continued)
Explore More
1. Circles come in different sizes, but all circles share the same shape.
Given any two circles, you can enlarge or reduce one circle on a
photocopy machine so that it matches the other.
Believe it or not, all parabolas possess the same property: Given any
two, it’s possible to enlarge or reduce one parabola on a photocopy
machine so that it matches the other.
Not convinced? Open the sketch Scale.gsp in the Parabola folder.
Two parabolas appear onscreen—one red, one blue. Drag the unit
point “1” on the x-axis. The blue parabola will stay in place, but the
red parabola will adjust to the change of scale on the x- and y-axes.
Because the scaling on the two axes grows and shrinks in unison, the
equation of the red parabola (y = x2) doesn’t change. By dragging the
unit point, you should be able to make the two parabolas overlap.

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Chapter 2: Parabolas • 37

The Folded Rectangle Construction

Name(s):

With nothing more than a sheet of paper and a single point on the page,
you can create a parabola. No rulers and no measuring required!
Constructing a Physical Model
Preparation: You’ll need a rectangular or square piece of wax paper or
patty paper. If you don’t have these materials, use a plain sheet of paper.
If you’re working
in a class, have
members place A at
different distances
from the edge. If
you’re working
alone, do this
section twice—
once with A close
to the edge, once
with A farther
from the edge.

1. Mark a point A approximately one
inch from the bottom of the paper
and centered between the left and
right edges.

A

2. As shown below right, fold the paper
so that a point on the bottom edge
lands directly onto point A. Make a
sharp crease to keep a record of this
fold. Unfold the crease.
3. Fold the paper along a new crease
so that a different point on the
bottom edge lands on point A.
Unfold the crease and repeat
the process.

A

4. After you’ve made a dozen or so
creases, examine them to see if
you spot any emerging patterns.
Mathematicians
would describe
your set of creases
as an envelope
of creases.

5. Resume creasing the paper. Gradually, you should see a
well-outlined curve appear. Be patient—it may take a little while.
6. Discuss what you see with your classmates and compare their folded
curves to yours. If you’re doing this activity alone, fold a second sheet
of paper with point A farther from the bottom edge.
Questions
Q1

The creases on your paper seem to form the outline of a parabola.
Where do its focus and directrix appear to be?

Q2

If you were to move point A closer to the bottom edge of the paper
and fold another curve, how do you think its shape would compare
to the first curve?

38 • Chapter 2: Parabolas

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

The Folded Rectangle Construction (continued)
Constructing a Sketchpad Model
Fold and unfold. Fold and unfold. Creasing your paper takes some work.
Folding one or two sheets is fun, but what would happen if you wanted to
continue testing many different locations for point A? You’d need to keep
starting over with fresh paper, folding new sets of creases.
Sketchpad can streamline your work. With just one set of creases, you
can drag point A to new locations and watch the crease lines adjust
themselves instantaneously.
7. Open a new sketch. Use the
Line tool to draw a horizontal
line near the bottom of the
screen. This line represents
the bottom edge of the paper.
8. Draw a point A above the line,
roughly centered between the
left and right edges of the screen.

A

crease

B

9. Construct a point B on the
horizontal line.

10. Construct the “crease” formed when point B is folded onto point A.
11. Drag point B along its line. If you constructed your crease line
correctly, it should adjust to the new locations of point B.
If you don’t want
your traces to fade,
be sure the Fade
Traces Over Time
box is unchecked
on the Color panel
of the Preferences
dialog box.

12. Select the crease line and choose Trace Line from the Display menu.
13. Drag point B along the horizontal line to create a collection of
crease lines.
14. Drag point A to a different location, then, if necessary, choose
Erase Traces from the Display menu.
15. Drag point B to create another collection of crease lines.
Retracing creases for each location of point A is certainly faster than
folding paper. But we can do better. Ideally, your crease lines should
relocate automatically as you drag point A. Sketchpad’s powerful
Locus command makes this possible.
16. Turn tracing off for your original crease line by selecting it and once
again choosing Trace Line from the Display menu.
17. Now select your crease line and point B. Choose Locus from the
Construct menu. An entire set of creases will appear: the locus of
crease locations as point B moves along its path. If you drag point A,
you’ll see that the crease lines readjust automatically.

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Chapter 2: Parabolas • 39

The Folded Rectangle Construction (continued)
Questions
Q3

How does the appearance of the curve change as you move point A
closer to the horizontal line?

Q4

How does the appearance of the curve change as you move point A
away from the horizontal line?

Playing Detective
Each crease line on your paper touches the parabola at exactly one point.
Another way of saying this is that each crease is tangent to the parabola.
By engaging in some detective work, you can locate these tangency points
and use them to construct just the parabola without its creases.
18. Open the sketch Folded Rectangle.gsp in the Parabola folder. You’ll
see a thick crease line and its locus already in place.
19. Drag point B and notice that the crease line remains tangent to the
parabola. The exact point of tangency lies at the intersection of two
lines—the crease line and another line not shown here. Construct this
line in your sketch as well as the point of tangency, point D.
Select the locus
and make its width
thicker so that it’s
easier to see.

20. Select point D and point B and choose Locus from the Construct
menu. If you’ve constructed point D correctly, you should see a
curve appear precisely in the white space bordered by the creases.
How to Prove It
The Folded Rectangle construction seems to generate parabolas. Can you
prove that it does? Try developing a proof on your own or work through
the following steps and questions.
The picture below should resemble your Sketchpad construction.
Line EF (the perpendicular bisector of segment AB) represents the
crease line formed when point B is folded onto point A. Point D sits
on the curve itself.
G
F

D

A
C

E

40 • Chapter 2: Parabolas

B

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

The Folded Rectangle Construction (continued)
Questions

Remember, a
parabola is the set of
points equidistant
from a fixed point
(the focus) and
a fixed line
(the directrix).

Q5

Assuming point D traces a parabola, which two segments must you
prove equal in length?

Q6

Use a triangle congruence theorem to prove that jACD m jBCD.

Q7

Use the distance definition of a parabola and the result from Q6 to
prove that point D traces a parabola.

Explore More
1. Open the sketch Tangent Circle.gsp in the Parabola folder. You’ll see
a circle with center at point C that passes through point A and is
tangent to a line at point B. Drag point B. Why does point C trace a
parabola?
2. A parabola can be described as an ellipse with one focal point
at infinity.
Open the sketch Conic Connection.gsp in the Parabola folder. You’ll
see the ellipse and circle from the Folded Circle construction. Press the
send focal point to “infinity” button. Point A—a focal point of the ellipse
and the center of the circle—will travel far off the screen.
As point A moves, notice how this affects the appearance of the visible
portion of the circle. Compare what you see to the Folded Rectangle
construction. In what ways do they appear similar?
3. Use the illustration from your parabola proof to show that
⇧GDF = ⇧ADC. The sketch Headlights.gsp in the Parabola folder
illustrates a nice consequence of this result.

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Chapter 2: Parabolas • 41

The Expanding Circle Construction
In response to those
who advised him to
take life easy, Ibn
Sina is said to have
replied, “I prefer a
short life with width
to a narrow one with
length.” He died
at the age of 58.

Name(s):

In this activity, you’ll explore a little-known parabola construction
from the tenth century. The method originates from Ibn Sina, a
jack-of-all-trades who was a physician, philosopher, mathematician,
and astronomer!
Constructing a Sketchpad Model
1. Open a new sketch. Choose
Show Grid from the Graph menu.
Then choose Hide Grid to remove
the grid lines while keeping the
x- and y-axes.

G

D

H

6

4

C
2

2. Label the origin as point A.
3. Choose the Compass tool. Click
on the y-axis above the origin
(point C) and then below the origin
(point B). This creates a circle with
center at point C passing through point B.

A
-5

E

F
-2

5

B

4. Construct point D, the intersection of the circle and the positive y-axis.
5. Construct points E and F, the intersections of the circle and the x-axis.
6. Construct lines through points E and F perpendicular to the x-axis.
7. Construct a line through point D perpendicular to the y-axis.
8. Construct points G and H, the intersections of the three newly
created lines.
If you don’t want
your traces to fade,
be sure the Fade
Traces Over Time
box is unchecked
on the Color panel
of the Preferences
dialog box.

9. Select points G and H and choose Trace Intersections from the
Display menu. Drag point C up and down the y-axis and observe
the curve traced by points G and H.
The curve you see is the locus of points G and H as point C travels
along the y-axis.
10. Drag point B to a new location, but keep it below the origin. Then, if
necessary, choose Erase Traces from the Display menu to erase your
previous curve. Trace several new curves, each time changing the
location of point B.
For every new location of point B, you need to retrace your curve.
Ideally, your parabola should adjust automatically as you drag point B.
Sketchpad’s powerful Locus command makes this possible.

42 • Chapter 2: Parabolas

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

The Expanding Circle Construction (continued)
11. Turn tracing off for points G and H by selecting them and once again
choosing Trace Intersections from the Display menu.
12. Now select points G and C. Choose Locus from the Construct menu.
Do this again for points H and C. You’ll form an entire curve: the
locus of points G and H. Drag point B to vary the shape of the curve.
Questions
Q1

As you drag point B, which features of the curve stay the same?
Which features change?

Q2

The creator of this technique, Ibn Sina, didn’t, of course, have
Sketchpad available to him in the tenth century! How would
this construction be different if you used a compass and
straightedge instead?

The Geometric Mean
It certainly looks like the Expanding Circle method draws parabolas,
but to prove why, you’ll need to know a little about geometric means.
ab .

The geometric mean x of two numbers, a and b, is equal to
Equivalently, x2 = ab.
Thus the geometric mean of 4 and 9 is
(4)(9) = 6

It’s possible to determine the geometric mean of two numbers
geometrically rather than algebraically. Specifically, if two segments have
lengths a and b, we can construct—without measuring—a third segment of
length a b .
13. Open the sketch Geometric Mean.gsp in
the Parabola folder.
You’ll see a circle whose diameter
consists of two segments with lengths
a and b laid side to side. A chord
perpendicular to the diameter is
split into equal segments of length x.

x
a

b
x

14. Use Sketchpad’s calculator to compute the
geometric mean of lengths a and b. Compare this value to x.

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Chapter 2: Parabolas • 43

The Expanding Circle Construction (continued)
Questions
Q3

The second page of Geometric Mean.gsp outlines a proof showing
that x is the geometric mean of a and b. Complete the proof.

How to Prove It
With your knowledge of geometric means, you can now prove that
points G and H of the Expanding Circle construction trace a parabola.
Since the location of point H changes as the circle grows and shrinks,
it’s labeled below as (x, y), using variables as coordinates. To make
things more concrete, we’ll assume AB = 3.
G

H = (x, y)

D
6

4

2

C
A = (0, 0)

E
-5

F
5

-2

B (0, -3)

Questions
The questions that follow provide a step-by-step guided proof. You can
answer them or first write your own proof without any hints.
Q4

Fill in the lengths of the following segments in terms of x and y:
AF =
AD =

Use your knowledge of geometric means to write an equation relating
the lengths of AB, AF , and AD . Is this the equation of a parabola?
Q6 Give an argument to explain why point G also traces a parabola.
Q5

Q7

Rewrite your proof, this time making it more general. Let AB = s.

Explore More
1. Open the sketch Right Angle.gsp in the Parabola folder. Angle DEB is
constructed to be a right angle. Drag point E and observe the trace of
point G and its reflection G . Explain why this sketch is essentially the
same as the Expanding Circle construction.

44 • Chapter 2: Parabolas

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Parabola Projects
The projects below extend your parabola knowledge in new directions
and are ideal for in-class presentations.
1. Build your own physical model of the Sliding Ruler construction.
2. Create a custom tool that automatically builds a parabola given a
directrix and a focal point.
3. In the Folded Rectangle construction, you built the crease line formed
when point B is folded onto point A by constructing the perpendicular
bisector of segment AB. Imagine now that Sketchpad’s Midpoint and
Perpendicular Line commands are broken. How can you construct
the crease line without them?
The picture below offers one possibility. It’s from the seventeenthcentury mathematician Frans van Schooten and shows a rhombus
EBFA with a slotted rod passing through points E and F.
Open the sketch Rhombus.gsp in the Parabola folder and experiment
with the model. How is this model similar to the Folded Rectangle
construction? What purpose does the rhombus serve?
E
B

A
F
D

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Chapter 2: Parabolas • 45

3
Hyperbolas

CHAPTER OVERVIEW
The shadows cast on a wall by a
lamp with a cylindrical shade. The
paths of comets that enter the inner
solar system and then leave forever.
The mirrors in many reflecting
telescopes. All of these are examples
of hyperbolic curves.
The picture at right shows a double
cone being sliced by a plane. The
cross section (the “conic section”)
is a hyperbola.
After an introduction to the
distance definition of a hyperbola
in activity I, a variety of hyperbola
construction techniques are
presented in activities II and III.
In total, you’ll find seven methods
to sample. Open the multi-page
sketch Hyperbola Tour.gsp for a
handy slide-show overview.

When a plane cuts both halves of a
double-cone, the result is a hyperbola.
The plane need not be parallel to the
double-cone’s axis.

I. Introducing the Hyperbola
Gold coins await the treasure seekers who successfully apply the
distance definition of a hyperbola.
II. The Concentric Circles Construction
The distance definition of a hyperbola serves as a springboard for
two hyperbola constructions: one built from concentric circles, the
other from a rotating ruler.
III. Hyperbola Projects
Round out the chapter with more hyperbola goodies.

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Chapter 3: Hyperbolas • 49

Introducing the Hyperbola

Name(s):

What does it take to find buried treasure? A map, a shovel—and would
you believe a hyperbola or two? Yes, with nothing more than two
hyperbolas, you can track down some hidden gold. You’ll be on your
way once you learn the definition of these curves.
What Is a Hyperbola?
Here is the geometric definition of a hyperbola:
A hyperbola is the set of points P such that the difference of
the distances from P to two fixed points (the foci) is constant.
If you open the sketch Hyperbola.gsp in the Hyperbola folder, you’ll
see a hyperbola along with its foci, F1 and F2. Every hyperbola consists
of two separate branches. Point P currently sits on the left branch of
the hyperbola.
For an ellipse, the sum of the distances from every point on the curve to
the two foci remains constant. But for a hyperbola, it is the difference of the
two distances that remains constant.
Of course, the difference of two numbers is either a positive or a negative
number, depending on the order of subtraction. If you move point P along
the left branch of the hyperbola, then drag it onto the right branch, you’ll
see that the value of PF2 – PF1 switches from positive to negative.
Dragging either
focal point to a new
location will change
the value of this
constant difference.

In a hyperbola, it is the absolute value of PF2 – PF1 that remains constant.
Every hyperbola also has two lines associated with it: the asymptotes. Press
the Show Asymptotes button to view them. The two branches of the
hyperbola approach these lines but never touch them. Said another way,
the distance between the branches and the asymptotes approaches zero,
but never reaches it.

50 • Chapter 3: Hyperbolas

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Introducing the Hyperbola (continued)
ʻXʼ Marks the Spot
Legend has it that the island of Keypress contains a buried treasure chest
of gold. After years of searching, you find the following note:
My treasure is two miles farther away from the giant boulder
(point B) than the lighthouse (point L). It’s also one mile
farther away from the cave (point C ) than the jail (point J).

Open the sketch Treasure.gsp in the Hyperbola folder to view the
landmarks in the note. Can you pinpoint the exact spot where the
treasure is buried?
As help, the sketch contains two hyperbolas: one with foci at B and L,
the other with foci at C and J. You can change the constant difference
associated with each hyperbola by dragging the segments that sit along
the left edge of the sketch.
Explain your method and why it works.

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Chapter 3: Hyperbolas • 51

The Concentric Circles Construction

Name(s):

Circles that are concentric share the same center. In this activity, you’ll
use two sets of concentric circles to draw a hyperbola by hand. You’ll then
transfer this technique to Sketchpad to draw a hyperbola whose shape
and size can be adjusted by just dragging your mouse.
Sketching Hyperbolas by Hand
The illustration that follows shows two sets of concentric circles. One set
of circles is centered at point F1, the other at point F2. For each set, the radii
of the circles increase by 1’s, from 1 unit all the way up to 7 units.
Points F1 and F2 are the foci of an infinite number of hyperbolas, but only
two that we’re interested in: one hyperbola that passes through point A
and another that passes through point B.
Q1

How many units apart are points A and F1? How many units apart
are points A and F2? What is the numerical value of AF1 – AF2?

Q2

Locate and mark at least 16 points that sit on either branch of the
hyperbola passing through point A. Explain how you found them.

Q3

Locate and mark at least 16 points that sit on either branch of the
hyperbola passing through point B. (Use a different colored pen or
pencil if possible.) Explain how you found them.

Q4

Using the points you found as guidelines, sketch the two hyperbolas.

A
B
F1

52 • Chapter 3: Hyperbolas

F2

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

The Concentric Circles Construction (continued)
By dropping two stones into a peaceful pond, you can create an animated
version of the concentric circles. Open the sketch Ripples.gsp in the
Hyperbola folder and press the Animate button to view some ripples and
their accompanying hyperbolas.
Examining a Sketchpad Model
Now that you’ve drawn some hyperbolas by hand, it’s time to examine a
dynamic one: a hyperbola that changes shape as its parts are dragged.
Open the sketch Concentric
Circles.gsp in the Hyperbola
folder. You’ll see two circles—
one red, one blue. The sizes of
these two circles are controlled
by the segments at the top of
the screen. The red circle has
radius AC and center F1, the
blue circle has radius BC and
center F2.

A

B

C

F1

F2

To operate this model, drag
point C. As you do, the radii
will adjust to remain equal to
AC and BC. At the same time,
you’ll be tracing the intersection
points of the two circles.
Drag point C to the right of point B, then back to the left of point A.
Questions
Q5

As you drag point C, the radii of both circles change lengths. Still,
there is a relationship that exists between the two radii whenever
point C is to the right of point B or to the left of point A. What is it?

Q6

Explain why the intersection points of the two circles trace
a hyperbola.

Q7

Select your two onscreen circles, choose Trace Circles from the
Display menu, then drag point C.
Based on this experiment, describe the similarities between your
Sketchpad construction and the concentric circles technique.

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Chapter 3: Hyperbolas • 53

The Concentric Circles Construction (continued)
The Rotating Ruler Method
Open the sketch Rotating Ruler.gsp in the Hyperbola folder. You’ll see
the model below.
The rectangle represents a ruler that rotates around the fixed point F2.
A string is attached from point F1 to the corner of the ruler (point B).
The string is held taut against the edge of the ruler by a pencil at point A.
Pulling the pencil up along the edge of the ruler causes the ruler to rotate
while point A traces a piece of a hyperbola.
To operate the Sketchpad model, drag point B.
F1

F2

A
B

How to Prove It
Can you prove that the Rotating Ruler draws hyperbolas? Try developing
a proof on your own or work through the following questions.
Questions
Q8

If the pencil at point A does indeed trace a hyperbola with foci at
F1 and F2, then what value must you prove constant?

Q9

Explain why the following equality holds:
(BA + AF2 ) – (BA + AF1 ) = constant

Q10

Complete the proof.

54 • Chapter 3: Hyperbolas

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Hyperbola Projects
The projects below extend your hyperbola knowledge in new directions
and are ideal for in-class presentations.
1. In the Folded Circle Construction (Chapter 1) you picked a point B
within a circle and then folded the circle repeatedly so that points
on its circumference landed on B. The outline of creases formed an
ellipse. If you’ve saved your sketch from that activity, open it. If not,
open Folded Circle Revisited.gsp in the Hyperbola folder.

Hint: In the Folded
Circle Construction,
you connected
points A and C
with a segment.
Try something
similar here.

What happens when point B sits outside the circle? Drag point B to
find out. (You might want to model this construction with paper
also—draw a circle on a sheet of notebook paper, mark a point B
outside the circle, then fold point B repeatedly onto different points
along the circumference.)
Can you prove that this modified construction generates hyperbolas?
To do so, you’ll need to find which point along each crease line is
tangent to the hyperbola.
2. In Project 1 above, you built the crease line formed when point C is
folded onto point B by constructing the perpendicular bisector of
segment BC. Imagine now that Sketchpad’s Perpendicular Line and
Midpoint commands are broken. How can you construct the crease
line without them?
The linkage below from seventeenth-century mathematician Frans
van Schooten offers one possibility. It shows a rhombus FBGC with a
slotted rod passing through points F and G.
What purpose does the rhombus serve? How is this model similar to
that in Project 1?
F
C

A

B
G

E

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Chapter 3: Hyperbolas • 55

Hyperbola Projects (continued)
3. Open the sketch van Schooten.gsp in the Hyperbola folder. You’ll see
a working model of the linkage below from the seventeenth-century
mathematician Frans van Schooten. Drag point C and observe the
trace of point E.
Can you prove that point E traces a hyperbola? The second page of
the sketch contains some hints to get you started.

C

A

B

F
E

4. The sketch Tangent Circles.gsp in the Hyperbola folder shows a red
circle c3 that’s simultaneously tangent to circles c1 and c2. Press the
Animate button and observe the path of point C, the center of circle c3.
Can you prove that C traces a hyperbola?
5. Rectangular hyperbolas are of the form xy = c, where c is a constant.
Open the multi-page sketch Area.gsp in the Hyperbola folder to learn
about some applications of these curves.

56 • Chapter 3: Hyperbolas

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

4
Optimization

CHAPTER OVERVIEW
What do a burning tent, a circular
swimming pool, and a cowgirl
have in common? Nothing, perhaps,
except that all three appear in this
chapter in geometric optimization
problems. Can you find a speedy
path to your burning tent before
only ashes remain? Can you swim
to your friend with only a minimum
of effort? Can you help a cowgirl
lead her horse to food and water by
plotting the shortest riding distance?
Normally, the topic of optimization
doesn’t arise until calculus, and
there it is treated algebraically.
Calculus is a great tool, but it’s not
Ellipses aren’t generally considered
standard firefighting tools, but perhaps
the only way to solve optimization
they should be . . .
problems. There’s only one
prerequisite for this chapter—the
Pins-and-String Construction from Chapter One. With that alone, you’re
ready to approach optimization problems from a purely geometric
perspective. Follow the activities in the order listed—they’re sequenced
to build on each other.
I. The Burning Tent Problem
With your camping tent on fire, it’s mathematics to the rescue as you
determine the optimal location for collecting some much-needed water.
II. The Swimming Pool Problem
A lazy day in a swimming pool turns mathematical when you agree to
buy the next round of ice teas for a friend. Can you find the shortest
distance to paddle without breaking a sweat?
III. An Optimization Project
Apply your optimization knowledge to this cowgirl conundrum.

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Chapter 4: Optimization • 59

The Burning Tent Problem

Name(s):

Consider the following situation:
Ah, the great outdoors. Camping, the fresh air, the starry night sky,
and—fire! Your tent is ablaze! Fortunately, you (at point Y ) have a
bucket in hand. You decide to run to the river’s edge, fill your bucket,
and race to your tent (point T ) to douse the flames.
Where along the river should you head in order to minimize your
total running distance? The picture below shows one possible
location—point P—you might run to.

Yes, we admit it:
If our tent were
really on fire, we
wouldn’t stop to
do math either!

This is an example of an optimization problem. You’re trying to optimize
the distance YP + PT to make it as small as possible.
This problem can be solved algebraically, but it becomes very messy.
So instead, you’ll investigate the situation geometrically, first with string,
then with Sketchpad.
Constructing a Physical Model
Text

Preparation: You’ll need a sheet of paper (larger is better), some string,
a pencil, a ruler, and tape.
1. Draw a long line on your paper to represent the river’s edge. Leave
space below the line to represent the water. Draw two points, Y and T,
to represent you and the tent.
2. Cut a length of string and attach its ends with tape or thumbtacks to
points Y and T. If the string does not extend below the river’s edge
when pulled taut, cut a longer piece or relocate Y and T.
Questions
Q1

Without taking any measurements, use your string to find the optimal
location along the river to run. Explain your method.

60 • Chapter 4: Optimization

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

The Burning Tent Problem (continued)
The Burning Tent problem can be solved in several ways. The questions
that follow describe one especially interesting technique. As you answer
the questions, think about the similarities between your method for
finding the optimal river location and the one presented here.
Before you begin, make sure your string is set up as before, with its ends
attached to points Y and T. The string should be long enough to extend
below the river’s edge when pulled taut.
Q2

Use a pencil to pull the string taut so that the pencil point sits directly
on the river’s edge. Call this point P. Point P is one location along the
river you might run to. It’s probably not the best location, but it will
serve as an initial guess.
Use your string to find another point along the river’s edge equivalent
to P in total running distance. Describe how you found the point.

Q3

Ignore, for a moment, the context of this problem. Use your string
and a pencil to draw all points, whether on land or in the water,
equivalent to P in total distance. Describe this set of points.
What curve have you drawn?

Q4

Use the curve from Q3 to identify two intervals on the river’s edge:
those locations whose total running distance is less than that
of P’s and those locations whose total running distance is greater
than that of P’s.
Explain how you found these intervals.

Q5

How should you proceed in order to find the optimal location
along the river?

Constructing a Sketchpad Model
The ideas behind the string solution to the Burning Tent problem can be
applied to a corresponding Sketchpad model. Here’s how:

You may have built
such a tool if you
did Project 6 from
Ellipse Projects
(page 29). If so,
use it instead.

3. Open the sketch Burning Tent.gsp in the Optimization folder.
You’ll see “you” (point Y), the tent (point T), and a point P along
the river you might run to.
4. Choose Ellipse by Foci/Point from the Custom Tools menu in
the Toolbox. This tool takes any three points—F1, F2, and P—and
constructs an ellipse with foci at F1 and F2, passing through point P.

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Chapter 4: Optimization • 61

The Burning Tent Problem (continued)
Questions
Q6

Use the Ellipse by Foci/Point tool to find the optimal river location to
run to. What is the relationship between the ellipse and the river at the
optimal location?

Q7

Can there be more than one optimal location for a straight river?
Explain.

Q8

Draw a curvy river (by hand, if you prefer) for which there are two
optimal locations to run to. Include an ellipse in your drawing to
justify your answer.

Q9

Suppose that regardless of where you ran to along the river’s edge,
the total distance from you (point Y) to the river to the tent (point T)
was the same. What would such a river look like?

A Reflection Technique
The Burning Tent problem is quite old. It appears in the 1917 book
Amusements in Mathematics by the great puzzlemaster Henry Ernest
Dudeney, where it’s called the “Milkmaid Puzzle.”
Dudeney’s method for solving the problem is totally different from the
one you just used. Follow the steps below to understand his approach.
5. Open the sketch Reflection.gsp in the Optimization folder. You’ll see
you (point Y), the tent (point T), and an arbitrary point P along the
river you might run to.
6. Double-click the line representing the river’s edge to mark it as a
mirror line of reflection. The line should flash briefly to indicate that
it’s been marked.
7. Select point T using the Arrow tool. Then choose Reflect from the
Transform menu to reflect point T across the river’s edge. Label the
reflected point as T .
8. Connect point P to point T with a segment PT .
Questions
Q10

Explain why the following equality holds for any location of point P:
YP + PT = YP + PT

Q11

Based on the equality from Q10, explain how to find the optimal
location of point P. Then use the ellipse tool to see if it yields the
same result.

Q12

Solve this problem by reflecting point Y instead of point T.

62 • Chapter 4: Optimization

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

The Swimming Pool Problem

Name(s):

Now that you’ve put out the flames from the Burning Tent, how about
a well-earned rest? It’s just you, a friend, and a swimming pool. But
wouldn’t you know it—the ellipse from the Tent problem returns!
Read on . . .
Tea for Two
You (at point Y ) and your friend (at point F ) are floating on inflatable
lounge chairs in a circular swimming pool. Waiters are positioned all
around the edge of the pool, and it’s your turn to buy two iced teas.
You’ll need to paddle to the edge, buy the drinks, and deliver one to your
friend. Where along the pool’s edge should you paddle to in order to
minimize the total distance? The picture below shows one possible path.
P

F (friend)

Y (you)

Try changing the
location of points Y
and F. Press the
Reset button when
you’re done.

Open the first page of the sketch Swimming Pool.gsp in the
Optimization folder. Besides the swimming pool, you’ll see an ellipse
with foci at points Y and F that passes through an arbitrary point P on
the circle’s circumference. Drag point P. The ellipse adjusts itself, with
Y and F remaining the foci.

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Chapter 4: Optimization • 63

The Swimming Pool Problem (continued)
Questions
Q1

Notice that there are four positions of P where the ellipse is tangent
to the circle. As with the Burning Tent, these locations are special.
Use the four circles that follow to draw the four tangent ellipses.

F (friend)

Y (you)

F (friend)

Y (you)

F (friend)

F (friend)

Y (you)

Q2

Y (you)

Now open the second page of the sketch and drag point P. You’ll
see that four locations on the circle’s circumference—P2 through P5—
correspond to the points of tangency you drew above. The sketch
also includes two more points, P1 and P6, for reference.
Here, in no particular order, are descriptions of locations
P2 through P5:
• This location gives the shortest overall paddling distance.
• This location gives the longest overall paddling distance.
• This location isn’t the best, but it’s better than all nearby
locations on either side of it.
• This location isn’t the worst, but it’s worse than all nearby
locations on either side of it.
Match the descriptions to the locations. Explain how the
positions of the ellipse, relative to the pool, allowed you to
draw your conclusions.

Q3

From shortest overall paddling distance to longest, rank the four
locations P2, P3, P4, and P5.

64 • Chapter 4: Optimization

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

The Swimming Pool Problem (continued)
Q4

Below is a set of axes for graphing. The horizontal axis represents
positions along the edge of the pool. For convenience, six locations
are labeled—P1 through P6. (Imagine the pool’s edge cut and then
straightened into a segment.) The vertical axis represents the total
distance it takes to paddle to your friend.
Use the information from Q2 and Q3 to draw a rough graph of the
location along the edge versus total paddling distance. The graph
need not represent the actual distances; rather, strive to make the
shape of the graph and the relative heights at P1 through P6 as accurate
as possible.

Distance
to paddle

P1 P2

P3

P4

P5

P6

Location along the pool’s edge

Q5

Believe it or not, you’ve just done some calculus! We use the following
terms in calculus to describe points on a graph: absolute maximum,
absolute minimum, relative maximum, relative minimum.
Make an educated guess as to the meanings of these terms, then
match the terms to points P2 through P5.

Q6

If you open page 3 of Swimming Pool.gsp and press the Show Graph
button, you’ll see the graph from Q4. Experiment with its shape by
moving the locations of Y and F.
How should Y and F be positioned to make the entire graph
horizontal, or nearly so?

Exploring Conic Sections with The Geometer’s Sketchpad
© 2012 Key Curriculum Press

Chapter 4: Optimization • 65

An Optimization Project
The project below builds on your knowledge from the Burning Tent and
Swimming Pool problems.
The Cowgirl Problem
Open the sketch Cowgirl.gsp in the Optimization folder and consider the
following situation:
A cowgirl wants to give her horse some food and water before
returning to her tent. She starts at point C and decides to travel
first to the pasture, then to the river, and then back to her tent.
What path should she take to minimize her riding distance?

Pasture
Points A and B
are two possible
locations the
cowgirl might take
her horse to.

C (Cowgirl)

T (Tent)

A
B

River

Solve this problem in two ways: with ellipses (use the provided custom
tool) an