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Pythagoras Plugged In guides students through a variety of proofs and applications of the Pythagorean theorem. By constructing and dynamically manipulating figures, students visualize the theorem and gain insights that no static illustration can offer. In addition to the activities, Pythagoras Plugged In includes historical background on Pythagoras and the theorem, teacher’s notes, and an appendix on creating animated presentation sketches.

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

Year:
2003
Publisher:
Key Curriculum Press
Language:
english
Pages:
103
ISBN 10:
1559536497
File:
PDF, 704 KB
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®The Geometer’s Sketchpad, ®Dynamic Geometry, and ®Key Curriculum Press are
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Limited Reproduction Permission
© 2003 Key Curriculum Press. All rights reserved. Key Curriculum Press grants the
teacher who purchases Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
the right to reproduce activities and sample sketches for use by his or her own students.
Unauthorized copying of Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
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Pythagoras Plugged In CD-ROM
Key Curriculum Press guarantees that the Pythagoras Plugged In CD-ROM that accompanies
this book is free of defects in materials and workmanship. A defective CD-ROM will be
replaced free of charge if returned within 90 days of the purchase date. After 90 days, there
is a $10.00 replacement fee.

Key Curriculum Press
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Acknowledgments
I owe thanks to all the following and, I’m sure, more: Leslie Nielsen, Sarah
Block, Rob Berkelman, Greer Lleuad, William Finzer, Nicholas Jackiw,
Scott Steketee, Annie Fetter, Steven Chanan, Daniel Ditty, and others
associated with Key Curriculum Press; Nathan King for;  A Four-Piece
Dissection, Doris Schattschneider for Pappus’ Theorem, Daniel Scher for
The Law of Cosines, Marc Roth for a new way to think about The Similar
Triangle Proof, Bill Medigovich for lending me his Loomis, Michael Serra
for Tessellating Pythagoras and general inspiration, David Bennett for
Wrong-Way Squares and mathematical upbringing, Steven Rasmussen for
making it possible for me to write the book; my colleague David Louis
and my students at The Nueva School; and thanks especially to Leslie, Eli,
and Gus.
D. B.

Contents
Introduction .................................................................................................. vii
A Little History.......................................................................................... vii
What’s in This Book ................................................................................... ix
To the Teacher: Using the Activities and the CD-ROM ........................... x
Chapter 1: The Theorem .............................................................................. 1
A Right Triangle with Squares................................................................... 3
Chapter 2: Proofs and Demonstrations .................................................... 7
Square Areas on a Grid .............................................................................. 9
Pythagorean Puzzles ................................................................................. 11
The Translator Tool .................................................................................... 12
A Dissection ................................................................................................ 13
Wrong-Way Squares .................................................................................. 15
The Tilted Square Proof ............................................................................ 17
Behold! ........................................................................................................ 20
E. A. Coolidge’s Proof ............................................................................... 22
Ann Condit’s Proof ................................................................................... 24
Leonardo da Vinci’s Proof ........................................................................ 26
Presidential Pythagoras ............................................................................ 29
Perigal’s Proof ............................................................................................ 31
Euclid’s Proof, a.k.a. Pythagoras’ Pants .................................................. 33
A Four-Piece Dissection ............................................................................ 35
Tessellating Pythagoras ............................................................................ 36
A Dynamic Proof ....................................................................................... 38
A Coordinate Proof ................................................................................... 42
The Similar Triangle Proof ....................................................................... 44
Chapter 3: Special Cases and Variations ................................................ 47
The Isosceles Right Triangle ..................................................................... 49
The 30°-60°-90° Triangle ............................................................................ 52
The Square Root Spiral .............................................................................. 53
A Pythagorean Tree .................................................................................. 55
Unsquare Pythagoras ................................................................................ 56
Pappus’ Theorem ....................................................................................... 58
The Law of Cosines ................................................................................... 60

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Contents • v

Chapter 4: Problems ................................................................................... 61
Activity Notes ............................................................................................... 67
Appendix: Making a Presentation Sketch .............................................. 87
References .................................................................................................... 91

vi • Contents

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Introduction
The Pythagorean theorem is without question the most famous, and
probably the most important, theorem in all of mathematics. This
simple equation, a2 + b2 = c2, would probably be recognized by everyone
who ever took a high school math class, whether they remembered what
it meant or not.
The legs of a right
triangle are the two
shorter sides. The
hypotenuse is the
longest side.

As you may know, that equation is an
abstraction of a special relationship among
the three sides of any right triangle, a
relationship that would probably be
surprising if it weren’t so familiar. In
geometric terms, the theorem states that the
sum of the areas of squares constructed on
the legs of a right triangle is equal to the area
of the square constructed on the hypotenuse.

c

a
b

Applications of the theorem are
endless—there are countless problems in
which an engineer, a carpenter, a scientist, an
architect, or a craftsperson might need to find an unknown or
unmeasurable length, distance, height, altitude, etc. If you can set up the
problem so that the unknown length is any side of a right triangle whose
other two sides you do know or can measure, then you can find the length
you’re looking for. Beyond practical applications of the theorem itself, the
right triangle relationship pervades many branches of mathematics (most
notably trigonometry), mathematics that has made the technological age
in which we live possible.

A Little History
I’ve made some pretty lofty claims about the importance of the
Pythagorean theorem. So who was this Pythagoras person whose name
we associate with the theorem, and should he get all the credit? Actually,
a number of different cultures around the world discovered the right
triangle relationship—or at least specific cases to which it
applies—independently of one another, long before Pythagoras lived.
I had a history of mathematics professor who even refused to call it the
Pythagorean theorem—he called it “the Babylonian theorem” in honor of
the ancient civilization from which comes a stone tablet, at least 3500 years
old, that lists several sets of three numbers that satisfy the Pythagorean
theorem. Specific cases of the relationship have been recorded by
numerous cultures—Arabic, Chinese, Indian, European, and probably
others—both before and after Pythagoras’ time.
Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Introduction • vii

As you read this book, you’ll see that the simplicity of the Pythagorean
theorem is one thing that makes it so fascinating. The theorem is such a
basic means of describing the physical world that it’s even been presumed
that any advanced culture would know it. In the early part of this century,
when many thought there might be intelligent life on Mars or on the
Moon, it was proposed that some huge illustration of the Pythagorean
theorem be sculpted into Earth’s landscape. When noted mathematician
Carl Friedrich Gauss was a teenager, he proposed cutting rows of trees in
Siberia in the shape of a right triangle with squares on the sides. Others
have suggested digging canals in the Sahara desert and burning kerosene
in them to illuminate a bright sign to our Martian neighbors that we
Earthlings are here and that we are reasonably smart.
No one knows
whether Pythagoras
came upon the
theorem in his
studies or
discovered it himself
independently.

By now you’re probably really wondering how such a universal theorem
got its name. Pythagoras was a Greek philosopher who lived around
500 B.C.E. His claim to fame is that he proved the theorem. A proof usually
offers explanation for why a theorem is true and establishes its truth for
some general class of figure, rather than just specific cases. Again, just as
other cultures knew specific instances of the theorem, they may also have
understood how the theorem applies to any right triangle. Pythagoras
himself lived and traveled throughout the Mediterranean and Asia Minor,
studying and teaching in Egypt, Babylonia, and what are now Italy and
Greece. Babylonia was an important center of world commerce, and
during his time there Pythagoras had the opportunity to study with
Babylonian, Egyptian, Chinese, and Indian scholars who may have known
the theorem.

One reason for the
influence of
Pythagoras’ school
was its longevity.
Pythagoras, in
violation of the laws
of the time, allowed
and encouraged
women scholars in
his society. He
married one of his
students, and his
wife and daughters
kept the school
active long after
Pythagoras’ death.

On his return to Greece, Pythagoras established a school that has had
lasting influence on the study of mathematics. Pythagoras’ teacher, Thales,
had begun a tradition of systematically showing how a theorem derives
logically from basic principles and other theorems. Thales proved
theorems about angles, including the angle sum theorem for triangles,
upon which any proof of the Pythagorean theorem depends. This
tradition, advanced considerably by Pythagoras’ school, is considered one
of the most influential contributions to modern intellectual thought.
Ironically, nobody is sure what Pythagoras’ proof of the theorem was. It
may be the proof found in Euclid’s Elements (300 B.C.E.), or one of the
simpler proofs you’ll study in this book. Whatever his proof, and whether
or not it was original, Pythagoras’ contribution to mathematics was
indisputably great, and few would begrudge him the honor of having his
name attached to the theorem.
Proving the Pythagorean theorem has probably interested people for as
long as the relationship has been known. As you’ll see, many proofs of the
theorem have a puzzle-like quality that makes them a pleasure to discover
and rediscover. Renaissance artist and engineer Leonardo da Vinci came
up with an original proof of the theorem, as did past United States
President James Garfield. Many of the proofs in this book come from a

viii • Introduction

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

book by Elisha Scott Loomis called The Pythagorean Proposition, first
published in 1927, that contains more than 300 proofs! Loomis was a high
school and college math teacher who made his life’s work collecting and
deriving his own interesting proofs of the theorem.

What’s in This Book
In this book, you’ll explore the Pythagorean theorem from a number of
different perspectives and strengthen your understanding of what it
means and why it is true. You’ll also apply the theorem to solve problems.
Along the way, you’ll learn more about the theorem’s interesting past and
how it has fascinated men and women of many different cultures since the
beginning of recorded history.
First, you’ll be introduced to constructing right triangles and squares—the
basic elements of the theorem—with The Geometer’s Sketchpad. Then
you’ll do constructions that illustrate a variety of different types of proofs.
In some activities, you’ll be asked to follow and supply some of the
reasoning to complete the proofs. You’ll explore a variety of different
types of proofs, but you may also notice common threads that run
through many of them. If you experiment with enough proofs, you’ll see
that there can be no limit to their number. You may even come up with an
original one of your own.
Once you’ve experienced some proofs of the theorem, you’ll look at
special triangles, including the isosceles right triangle that may have led to
another very important discovery by Pythagoras. You’ll also generalize
the theorem for shapes other than squares. And you’ll explore Pappus’
theorem and the law of cosines—generalizations of the Pythagorean
theorem that work even for nonright triangles.
In Chapter 4: Problems, you’ll solve problems using the Pythagorean
theorem, and you’ll model some of these problems with Sketchpad.
Finally, there’s an optional appendix in which you can learn some of the
basics of creating presentation sketches. Presentation sketches let you
demonstrate your construction and explain your reasoning to a user
without having to be there.
Chances are you won’t do everything in this book. But I hope these
activities inspire you to explore the Pythagorean theorem in depth. It’s
certainly been fertile ground for many budding mathematicians, amateur
and professional.

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Introduction • ix

To the Teacher: Using the Activities and the CD-ROM
The preceding introduction is written for you and your students.
I encourage you to share some of the history of the Pythagorean theorem
with your students, either in discussion preceding the activities or by
photocopying the introduction for them. That the theorem has fascinated
so many people for so many centuries may motivate your students to try
to learn more about what all the fuss is about.
The Activities
Some of the figures students are asked to construct in this book get
complicated, and all the activities assume this is not a student’s first
experience with Sketchpad. Students should be introduced to the program
more slowly than these activities do. The Guided Tours in the Learning
Guide that comes with Sketchpad are a good place to start.
Most of the activities in this book start with a short introduction followed
by a section titled Sketch and Investigate. In this section, students are
given the steps for constructing a figure that demonstrates the theorem.
For the most part, these steps are given in geometric terms, without
detailed instructions about the program’s user interface. Some of the
comments in the sidebars help students find where commands are in
Sketchpad’s menus. If students get stuck, make sure they look to the
sidebars for hints. Interspersed with these construction steps are some
questions to answer about the construction that preview ideas used in the
proof. The questions are indicated with the letter “Q” preceding the
question number. In some activities, the questions continue into the next
section, titled Prove. Here students get down to explaining the main ideas
of the proof.
The place to start in this book is with the activities in the section titled The
Theorem. Depending on students’ Sketchpad experience, this could take
one or two class periods. It’s very important in all the activities that
students be able to construct right triangles and squares. And you’ll want
students to be able to use a custom tool to construct squares, too.
Most of the activities in Chapter 2: Proofs and Demonstrations could
probably be done in one class period each, assuming written proofs were
left for homework. But unless you plan to spend a whole term on the
theorem, you’re obviously not going to have students do them all. A more
realistic option would be to assign groups or pairs of students different
activities. Give them a class period to get started on an activity, then if
possible, a longer period of time during which they can work on it outside
of class. Then save a few minutes of several class periods for student
demonstrations. It’s worthwhile for students to see several different
proofs of the theorem, even if it’s impractical to ask them to do many.
x • Introduction

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

The constructions are of varying difficulty and each activity is rated in the
activity notes as beginner, intermediate, or advanced. Not coincidentally,
how difficult it is to understand the proof tends to be proportional to the
difficulty of the construction.
It’s up to you to decide how much emphasis to give the formal proofs. The
constructions and investigation questions themselves are meant to guide
students to most of the insights behind the proofs, but typically there’s
more explanation required for a rigorous proof. The proofs also provide
opportunities to apply congruence and transformation axioms as well as,
in many cases, algebra. Nevertheless, some of the harder proofs may be
too much for many students. You should make sure you try the proofs
yourself to determine their appropriateness for your students. In my
proofs in the activity notes, I tried to err on the side of too much detail,
thinking that might be safer than taking too much for granted. In general,
I’d expect less detail from my students.
One option to consider is to be selfish and keep the book to yourself for a
year. I think you, as a math teacher, are going to get a kick out of this
book. And it may serve your students better if you take time to familiarize
yourself with it and try out some of the activities on your students as
teacher-led demonstrations. That should give you ideas about which
activities would go over best with students, as well as ideas for how best
to let students use the activities.
The CD-ROM
The CD-ROM contains sketches for Chapters 1 and 4 and folders for
Chapters 2 and 3. There’s also a sketch showing various stages of the
Appendix activity. Assuming your students do the constructions
themselves, these sketches are only for reference. Use them to check if you
or your students’ constructions came out the way they were supposed to.
There’s also a custom tool for constructing a square in the Chapter 1
sketch. Again, if students make their own tool, they won’t need this one.
The one exception is the activity A Four-Piece Dissection. This activity
requires students to start with a pre-made sketch.
An additional folder titled Presentations contains presentation
sketches—sketches with action buttons that enable users to step through
dynamic, visually based demonstrations of the theorem without doing
any constructions. The presentation sketches also correspond to some of
the activities. They’re a bit more razzle-dazzle than what students
construct in the activities, and they’re a means for quickly demonstrating
proofs students don’t get a chance to do themselves. But they’re no
substitute for the insight gained by struggling through a construction.
The Chapter 4 sketch contains sketches of scale drawings and models of
the problems in that chapter. Like the sketches for Chapters 1–3, these are
meant to serve primarily as examples. Students should try to construct
Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Introduction • xi

their own models and scale drawings. There are, of course, any number of
ways to model a given problem, so you needn’t be reluctant to share these
examples with students to help them learn how to do their own. Be sure to
read the “read me” file on the CD for an up-to-date description of the
CD’s contents.

xii • Introduction

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

The Theorem

The Pythagorean theorem, stated geometrically, says the sum of the
areas of the squares on the two legs of a right triangle is equal to the
area of the square on the hypotenuse.

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Chapter 1: The Theorem • 1

A Right Triangle
with Squares

Name(s):

The basics for exploring the Pythagorean theorem are that you need to be
able to construct a right triangle and squares. There are many ways to
construct both these shapes. Here you’ll learn one way for each.
1. Construct segment AB.
Select the segment
and the point,
then choose
Perpendicular
Line from the
Construct menu.

2. Construct a line
perpendicular to sAB
through B.
3. Construct sAC, where
point C is on the
perpendicular line.

This is just one of
many ways to
construct a right
triangle. Can you
come up with
others?

The Text tool
Show or hide an
object’s label by
clicking on the
object with the Text
tool. Double-click on
the label to edit it.
The hand turns dark
when positioned
over an object. The
letter “A” appears
when it’s positioned
over a label.

A

B

C

4. Construct sBC and hide the line.
Experiment with dragging each of the three vertices of the triangle. You’ll
find that they behave differently because they’re constructed with
different constraints.
Q1

Does your triangle always stay a right triangle? What step in your
construction guarantees this? Can your right triangle be every
possible size or shape?

Q2

Which two points behave the same? When you drag either of them,
how does the triangle change? Which point or points change the
shape of the triangle when dragged?

5. Relabel the sides as a, b, and c, with a the shortest leg, b the longer leg,
and c the hypotenuse. Relabel the vertices opposite these sides as A, B,
and C, respectively.
The diagrams in this book show the right triangle positioned and labeled
in the two ways shown at the top of the next page. Practice manipulating
and labeling your triangle so it matches the two diagrams.

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Chapter 1: The Theorem • 3

A Right Triangle with Squares (continued)
The hypotenuse in a
right triangle is
opposite the right
angle. In this book,
the hypotenuse will
usually be labeled c,
and the right angle
vertex opposite it will
be labeled C.

C

B

C

b

a

c

a

b

A

B

c

A

Can you come up with other ways to construct a right triangle?
The Square
6. Construct segment AB.
To mark a point as
center, select it and
choose Mark
Center from the
Transform menu.
The Rotate
command is also in
the Transform menu.

B'

A'

7. Mark point A as center, then rotate the
segment and point B by 90°.
8. Mark point B´ as center and rotate segment
AB´ and point A by 90°.
9. Connect points A´ and B to complete
your square.

A

B

10. Select the four vertices in consecutive order
and choose Quadrilateral Interior from the Construct menu.

The Custom tool
Making a tool of your
construction will
save you the work of
going through the
whole construction
next time you
want a square.

11. Select the entire figure, then press and hold the Custom tools icon.
Choose Create New Tool from the Custom Tools menu. Name the
new tool Square.
Q3

How do different vertices behave when you drag them? Which
vertices are most constrained and which are least constrained?

Q4

Does your square always stay a square? What is it about your
construction that guarantees your square will stay a square?

4 • Chapter 1: The Theorem

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

A Right Triangle with Squares (continued)
12. Experiment with using your new Square
tool. Choose it from the Custom Tools
menu. Click, move the mouse from left to
right, and click again. Try to construct a
second square below the first, using the
same two points your tool started with, but
go from right to left.
It will be important for you to be able to
construct a square on either side of a segment
when you explore the Pythagorean theorem.

B'

A'

B

A

A Right Triangle with Squares on the Sides
You’re now ready to construct squares on the
sides of a right triangle.
If one of your
squares overlaps the
triangle, don’t worry:
Just undo and try
applying your tool to
the two triangle
vertices in the
opposite order.

A'

13. Construct a right triangle ABC, either from
scratch or by using a tool.
14. Use your Square tool on the endpoints of each side of the triangle.
Once you have a right
triangle with squares on the
sides, you have a geometric
illustration of the
Pythagorean theorem.
15. Measure the areas of each
of the interiors and the
lengths of each side.
16. Do a calculation on each
side length that gives you
a value equal to the area
of the corresponding
square.

To calculate with
measures, choose
Calculate from the
Measure menu. Click
on the measures in
the sketch to enter
them in a
calculation.

B'

B
c

a

C

b

A

17. Perform a calculation
with areas of two squares
that gives you a value
equal to the area of the
third square.
Q5

Drag different vertices of the triangle and observe the calculations.
Write an equation in terms of side lengths a, b, and c that describes the
relationship you observe.

Q6

Does what you’ve done in this activity qualify as a proof of the
Pythagorean theorem? Discuss this question with your classmates.

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Chapter 1: The Theorem • 5

A Right Triangle with Squares (continued)
The Converse
The converse of the Pythagorean theorem states that if the sum of the
squares of two sides of a triangle is equal to the square of the third side,
then the triangle is a right triangle. Follow the steps below to investigate
the converse.
18. Use the Segment tool to draw an
arbitrary (nonright) triangle.
19. Construct squares on the sides of this
triangle.
20. Measure the areas of the three squares
and calculate the sum of two of them.
21. Drag a vertex until the sum equals the
area of the third square.
Q7

What kind of triangle do you have
when the sum of the areas of the two
smaller squares is equal to the area of
the larger square? Measure an angle to
confirm your conjecture.

6 • Chapter 1: The Theorem

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Proofs and Demonstrations

Tâbit ibn Qorra (826–901) translated Euclid’s Elements into Arabic. This is
his translation of Euclid’s proof of the Pythagorean theorem. If you have
trouble following this proof, remember, read from right to left.

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Chapter 2: Proofs and Demonstrations • 7

Square Areas on a Grid

Name(s):

In this activity, you’ll find areas of squares on a square grid, including
“tilted” squares. The strategy you develop for finding these tilted square
areas can be applied to one proof of the Pythagorean theorem.
Sketch and Investigate
1. Open the sketch Squarefinder.gsp.
2. Drag point A or B to get a feel for
how the corners of the square snap
to the grid.
Note that when the yellow square is
tilted on the grid, a larger square
with dashed sides surrounds it. Pay
attention to when the measure of sAB is
a whole number and when it is not.
Q1

B

A

A perfect square is a number
whose square root is a whole
number. In this sketch, the yellow
square has whole-number side lengths when its sides are horizontal or
vertical (that is, when the square is not tilted). In these nontilted
squares, the area is a perfect square. Sketch six different perfect
squares on a piece of graph paper and label each square with its side
length and area.

3. Now drag point A or point B to make a tilted square whose area you
think is 5 square units. When you’re pretty sure you have the right
square (and not before!), press the Show Area Yellow Square button to
check yourself. If you were wrong, press the Hide Area Yellow Square
button and try again.
The square with area 5 has a side length of the square root of 5. The
measure in the sketch is a decimal approximation.
Q2

Sketch the square with area 5 on your graph paper. Label it with its
area and its side length expressed as a square root and a decimal
approximation, that is,

5 ≈ 2.24 .

Q3

Make five more tilted squares. For each, figure out the area, check
yourself with the Show button, then sketch the square on your graph
paper. Label each with the area and the side length.

Q4

Explain your strategy for finding the areas of tilted squares.

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Chapter 2: Proofs and Demonstrations • 9

Square Areas on a Grid (continued)
Explore More
1. Write an expression for the tilted square
shown at right in terms of a and b. Call the
side length of the tilted square c, and write
an equation relating a, b, and c. Simplify the
equation as much as you can. Does this
equation apply even if the square is not on
a grid?
2. Some tilted squares on the grid have whole
number side lengths. Find as many of these
as you can.

b

a

a
b
b
a
a

b

3. Is it possible to make a tilted square on the grid with area 3? Explain.
4. Is it possible to make a square on the grid whose area is not a whole
number? Explain.

10 • Chapter 2: Proofs and Demonstrations

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Pythagorean Puzzles

Name(s):

Some demonstrations of the Pythagorean theorem are like puzzles. In
these puzzles, called dissections, you start with a picture that has several
pieces, which usually fit together to form one part of the Pythagorean
theorem (like a2 + b2). You cut out those pieces and rearrange them to form
the other part of the theorem (like c2). While simply rearranging pieces
doesn’t qualify as a proof of the theorem, these puzzles can lead to proofs
if you can explain why they work.
Sketch and Investigate
Open the sketch Pythagorean Puzzles.gsp. There are four puzzles, each
on its own page. Follow the directions in the sketch. Try all four puzzles.
Explore More
1. Write two expressions for the area of the square below left, one
in terms of a and b and the other in terms of a, b, and c. Set the
expressions equal and simplify to prove the Pythagorean theorem.

b

c

a

b

a

c

2. One expression for the area of the square above right is c2. Write
another expression in terms of a and b. Set the expressions equal and
simplify to prove the Pythagorean theorem.

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Chapter 2: Proofs and Demonstrations • 11

The Translator Tool

Name(s):

One common way of demonstrating the Pythagorean theorem is to cut out
pieces of the squares on the legs and rearrange them to fit in the square on
the hypotenuse. In this activity, you’ll create a custom tool that lets you do
the same kind of thing with Sketchpad—the tool will give you pieces that
you can move around freely. The tool uses a translation, which, in
geometry, means a transformation that creates an identical image by
sliding without turning or flipping. You’ll find this tool useful in a
number of activities in this book.
Sketch
1. Construct an arbitrary
quadrilateral ABCD and its
interior.
2. Construct a point E in a blank
area. This point is where you’ll
locate a copy of the quadrilateral
interior in the following steps.

E

C
B

3. Select, in order, point B and
point E and choose
Transform|Mark Vector.
4. Select only the quadrilateral
interior and choose
Transform|Translate and
translate by the marked vector.

D
A

5. Drag point E to observe that this image is free to move independently
of the original quadrilateral.
6. Drag any vertex of the original quadrilateral to observe that it and the
translated image are dynamically linked.
In the following steps, you’ll create a tool for this construction.
7. Select, in order, the original quadrilateral interior, point B, point E,
and the translated interior. Press and hold the Custom tools icon and
choose Create New Tool from the Custom Tools menu. Name the
tool Translator.
8. Practice using the Translator tool. Here’s how it should work: Click
on an interior that you wish to translate; click on one vertex of the
interior; click a third time anywhere in the sketch to create a translated
image with one vertex in that location.
9. Save your sketch so that you can use this tool in later activities.

12 • Chapter 2: Proofs and Demonstrations

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

A Dissection

Name(s):

One way to demonstrate the Pythagorean theorem is to cut the squares
into pieces so that the pieces of the squares on the legs can be arranged to
fit into the square on the hypotenuse (or vice versa). In this activity, you’ll
learn one such dissection.
Sketch and Investigate
1. Construct a right
triangle and squares on
the sides.
How can you find the
center of a square?

To construct a
parallel, select a
point to go through
and a straight
object to be
parallel to.

2. Construct the center of
the square on side b,
the larger of the two
legs.

B
c

3. Construct a line
through this center,
parallel to side c,
and another line
perpendicular to side c.

a
A

b

C

4. Construct points of
intersection of the lines
with the sides of the
square on side b.
Select the four
vertices and choose
Quadrilateral
Interior from the
Construct menu.

5. Construct the four
quadrilateral interiors of the
regions into which the square on
side b is divided. Give them
different shades or colors.

Select the lines, then
choose Hide Lines
from the Display
menu.

6. Hide the lines.
7. Construct the interior of the
square on a, the smallest leg.
The four pieces of the square on
side b, combined with the square on
side a, make five pieces that can be
rearranged to fit into the square on
side c. In the following steps, you’ll
make translated images of these
pieces that you can drag around
freely.

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

B
a

c
b

C

A

Chapter 2: Proofs and Demonstrations • 13

A Dissection (continued)
8. To create a translated
image of the square on
side a, choose the
Translator tool from the
Custom Tools menu, and
click on the following, in
order: the interior of the
square, a vertex of the
square, a blank area (to
construct the translated
copy). Repeat on the four
pieces in the square on
side b.

B
a

c
b

C

A

9. Now drag each piece into
the square on side c and see
if you can arrange them to
perfectly fill this square.
Q1

How does this demonstrate the Pythagorean theorem?

10. To confirm that this dissection works for other right triangles, change
the shape of your triangle and refit the pieces.

14 • Chapter 2: Proofs and Demonstrations

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Wrong-Way Squares

Name(s):

Typically, the Pythagorean theorem is illustrated with squares that lie
outside of a right triangle. But what if you construct squares the “wrong
way”? That is, what happens if one or more squares overlap the triangle?
In this investigation, you’ll see how constructing overlapping squares
divides your figure into pieces that can be rearranged to give a dissection
demonstration of the Pythagorean theorem.
Sketch and Investigate
1. Construct a right triangle.
If you’re using a tool,
you should be able
to control which way
the square goes by
the order in which
you drag from vertex
to vertex.

2. Construct a “wrong-way”
square on side a so that it
overlaps the triangle, and
construct a square on side b
that falls outside the triangle.

B

c

a

C

b

A

3. Construct a wrong-way
square on the hypotenuse,
overlapping the triangle.
4. The squares on sides a and b
are now divided into five
regions. Construct the points
where the squares intersect,
then construct the polygon
interiors of these regions.
Q1

Drag vertices of your triangle. Does this construction always divide
the square on side c into five pieces? Make note of special cases if you
encounter any.

5. One of the pieces in the square on side a also lies within the square on
side c, as does one of the pieces on side b. Identify the three regions
that don’t already lie in the square on side c. Use the Translator tool to
create translated images of these three pieces that you can drag
around. Hide the original pieces and fit the translated images in the
square on side c.
Q2

Explain how this demonstrates that the area of the square on side c is
equal to the sum of the areas of the squares on sides a and b.

6. Drag a vertex to change your triangle and refit the pieces to confirm
that this works for different-shape triangles.

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Chapter 2: Proofs and Demonstrations • 15

Wrong-Way Squares (continued)
Prove
Dissection demonstrations convincingly illustrate the truth of the
Pythagorean theorem, but they don’t provide any of the logic. You show
that you can rearrange the pieces, but you don’t explain why this works.
Why is it that you can move these pieces and they happen to fit within
other places? To explain this requires a logical argument proving that
each moved region is moved to a region congruent to it.
Q3

Can you find three
right triangles in the
figure that are
congruent to one
another?

Q4

In the labeled diagram,
which three regions moved?
Into which three regions did
each move?

B

K

J

C
A
The three regions that
I
moved (and the three
regions they moved into) are
H
L
all right triangles. Prove that
each region that moved is
congruent to the region it
G
was moved into. Remember,
to prove right triangles
F
E
D
congruent, you need
only show two pairs of
corresponding sides congruent (HL, LL) or one pair of sides and one
corresponding pair of nonright angles (HA, LA).

Explore More
In this activity, you constructed two wrong-way squares. Can you create a
different construction that yields a dissection by constructing just one
wrong-way square? How about by constructing two wrong-way squares
different from the ones in the activity? Hint: You may need to rotate one or
more pieces to fit them into the proper places when you move them.

16 • Chapter 2: Proofs and Demonstrations

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

The Tilted Square Proof

Name(s):

In this activity, you’ll construct a simple figure that you can use to prove
the Pythagorean theorem, either by moving figures or by using a little
algebra. The figure appears in very old texts and you may have even seen
it in decorative tile designs. Nobody knows, but it may have been the
proof known to Pythagoras, as it seems to have been known in China and
India and Pythagoras studied with Chinese and Indian scholars in
Babylonia.
Sketch and Investigate
1. Construct a square and the center of the square.
2. Construct a point on one side of the square.

Steps 1 and 2

Step 3

Steps 4 and 5

3. Mark the center of the square a center of rotation and rotate the point
on the side by 90°. Repeat on this rotated image until you have points
on all four sides.
4. Use the points to construct a tilted square inside the larger square.
5. Construct the interiors of the triangles in the corners of the larger
square.
Q1

Drag the point on the bottom side of the large square. Do the four
triangles appear to remain congruent? How do you know they’re
congruent? Answer in a paragraph.

Q2

What kind of shape is the empty region in the center of the figure? As
you change the figure, what changes about that shape? What doesn’t
change? Can you make it fill the original square? How small can you
make it?

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Chapter 2: Proofs and Demonstrations • 17

The Tilted Square Proof (continued)
6. Label the sides of at least one of the right triangles a, b, and c, where c
is the hypotenuse.
Q3

What’s the area of the tilted square inside the figure?

7. Now you’re going to move the triangles around. Use the Translator
tool to construct translated images of three of the four triangle
interiors. Hide the original three.
8. Arrange the triangles so that you have two
rectangles as shown.
Q4

What kinds of shapes are the two empty
regions now? What is the sum of their
areas?

9. Drag the bottom point to change the
figure and reposition the triangles to
demonstrate that this works for differentshape triangles.
Q5

Explain how this experiment demonstrates the Pythagorean theorem.

Prove
You can write an algebraic proof based on the demonstration you just did.
Here are some hints for planning the proof.

`

Q6

Show that the tilted square inside the figure really is a square, and
express its area in terms of c.

Q7

What’s the length of one side of the larger square (the whole figure)?
Write an expression for the area of this square in terms of a and b.

Q8

What’s the area of one triangle? What’s the sum of these four areas?

Q9

Write another expression for the area of the entire figure, this time in
terms of the areas of the triangles and the area of the tilted square.
This expression should be in terms of a, b, and c.

Q10

You now have two expressions representing the same thing. Write an
equation. You’re on your own from here.

18 • Chapter 2: Proofs and Demonstrations

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

The Tilted Square Proof (continued)
Explore More
Without too much work,
you can build onto your
already constructed
figure to make a nice
illustration of the proof.
In the figure at right, the
original figure was
reflected over its right
side. The Translator tool
was used to translate
triangles to existing
points so that they would be attached in place, forming white squares
with sides a and b, respectively. See if you can construct a figure like this
that can be manipulated dynamically so that all the triangles stay
congruent and the squares stay squares. By manipulating this figure, you
can study many interesting special cases, such as isosceles right triangles
or what happens when the triangles are made infinitely small.

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Chapter 2: Proofs and Demonstrations • 19

Behold!

The figure in the
Chinese text shows
a 3-4-5 right triangle,
but the figure works
for any right triangle.

Name(s):

The title of this activity comes from the text of the twelfth-century Hindu
scholar, Bhaskara. In fact, “Behold!” was the only text that accompanied a
figure demonstrating the Pythagorean theorem. Bhaskara must have felt
the figure spoke for itself! In this activity, you’ll construct this figure.
Perhaps it does speak for itself, but you can gain deeper understanding by
constructing the figure and working out a proof similar to the proof
outlined in the activity The Tilted Square Proof. Incidentally, this figure is
also found in an ancient Chinese text, making it another candidate for
being a proof known to Pythagoras.
Sketch and Investigate
1. Construct a square ABCD.
2. From point D, construct a
segment DE to side AB.

D

C

3. Construct a line parallel to sDE
through point B and lines
perpendicular to sDE through
points A and C.
4. You should have a small, tilted
square inside your original
square. Construct its vertices at
the points of intersection of the
lines, then hide the lines and sDE.

A

E

B

5. Construct the sides of the tilted square, then
construct interiors of the right triangles
surrounding it, as shown.
6. Drag point E and observe what happens to the
right triangles and the tilted square.
Q1

Write a paragraph about what you observe: Do
E
the right triangles stay right triangles? Does the
square stay a square? Can you make the interior
square fill the figure? If so, what kind of triangles do you get in this
case? Can you make the interior square disappear? What kind of
triangles will do this?

20 • Chapter 2: Proofs and Demonstrations

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Behold! (continued)
7. Using this figure for a dissection demonstration can be tricky. First, tilt
the original square so that the legs of the right triangles are horizontal
and vertical. The illustration below shows an outline into which the
pieces can be fit to demonstrate that c2 = a2 + b2. Note: Use the
Translator tool to get movable pieces, and try to form the shape
below right. Your pieces may overlap the dashed line.

E
a

b

Prove
This figure can be used to create an algebraic proof similar to the one you
might have done in the activity The Tilted Square Proof. A possible plan
follows. Read it if you want, or try it on your own.
Q2

Write an expression for the area of the whole square in terms of c.

Q3

Write an expression for the sum of the areas of the four right triangles
in terms of a and b.

Q4

The tricky part is writing an expression for the area of the tilted
square. (And if you want to be very thorough, you should prove it
really is a square.) The length of a side of this square can be written in
terms of a and b. Write an expression for the area of the tilted square
in terms of a and b.

Q5

You should now be able to write an equation involving a, b, and c.
You’re on your own from here.

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Chapter 2: Proofs and Demonstrations • 21

E. A. Coolidge’s Proof

Name(s):

This proof appears in Elisha Scott Loomis’s classic, The Pythagorean
Proposition, first published by the National Council of Teachers of
Mathematics in 1927. That book, now unfortunately out of print, contains
over 300 proofs of the Pythagorean theorem. Many are similar to one
another, and all the well-known proofs are found in Loomis’s book. The
following proof is similar to Bhaskara’s proof from the activity Behold!
Loomis’s credit reads, “credited to Miss E. A. Coolidge, a blind girl. See
Journal of Education, V. XXVIII, 1888, p. 17, 26th proof.”
Sketch and Investigate
Use a custom tool
to construct these
squares and the
square in step 4.

1. Construct a right triangle and squares on the sides.
2. Extend sHA to point A´ by translating sHA and point A by vector HA.
3. Construct a line through point B, perpendicular to sAA´, and construct
point of intersection K.
4. Construct square A´KLM.
5. Construct sBK, sGM, and sFL.
6. Hide line BK.
7. Construct the interiors of the four pieces in the square on the
hypotenuse.

Your figure so far
may look familiar.
It’s Bhaskara’s
dissection from the
activity Behold!

F

A'

M
E

G
K

B
L

D

a

c

C

b

J

A

H

In the following steps, you’ll construct identical pieces on one of the other
squares of the triangle.
22 • Chapter 2: Proofs and Demonstrations

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

E. A. Coolidge’s Proof (continued)
To mark a vector,
select two points, in
order from tail to tip,
and choose Mark
Vector from the
Transform menu. Be
sure to choose By
Marked Vector
when you translate.

8. Mark vector EJ and translate
the four sides and vertices of
BCDE by this marked vector.
9. The region on side b is now
a2 + b2. Use your Translator
tool to place copies of the
pieces of the square on the
hypotenuse in the a2 + b2
region on side b. If you choose
points carefully, you should
be able to attach all the pieces
to points on this region.
Q1

Q2

Change your triangle. Do the
corresponding pieces remain
congruent? Note that leg b
needs to be kept the longer leg
or the construction falls apart.
Describe the special case you
get just before your
construction falls apart.

F

M

G

B

E

L

D

A'

a

c

C

b

N

K

A
Q
P

J

B'

D'

C'

H

Explain why the pieces can fit in the region on side b the way they do.

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Chapter 2: Proofs and Demonstrations • 23

Ann Condit’s Proof

Name(s):

The proof in this activity also appears in Elisha Scott Loomis’s classic, The
Pythagorean Proposition. Ann Condit devised it in 1938 while a student at
Central Junior-Senior High School, South Bend, Indiana. Says Loomis,
“This 16-year-old girl has done what no great mathematician, Indian,
Greek, or modern, is ever reported to have done. It should be known as
the Ann Condit Proof.”
Sketch and Investigate
Be sure to use
point A as the
second control point
of your circle, rather
than constructing a
circle that’s “just
passing through”
point A.
This is a different
right triangle
construction than
you’ve done before.
What guarantees the
triangle is a right
triangle?

1. Construct a segment AB.
G

2. Construct the midpoint
D of this segment.

M
P

3. Construct circle DA.
4. Construct sBC and sAC,
where C is a point on the
circle, making right
triangle ABC.
5. Construct squares on the
sides of right triangle
ABC.

F

H
C

L
E

B

A

D

6. Construct midpoints L,
M, and N of the outside
sides of the squares.
These three
segments divide the
squares on the three
sides in half. Can
you see why?

7. Construct segments DL,
DM, and DN.

K

N

J

8. Construct sFG, fDC, and point P where fDC intersects sFG, then hide fDC
and replace it with sDP.
9. Construct the interiors of triangles DCF, DCG, and DBK.
This proof relates the areas of these shaded triangles to one another and to
the areas of the squares on the sides. Before you attempt the proof, you
might want to investigate what these relationships are.
10. Measure the areas of the triangles and drag point C around the top
half of the circle, back and forth between points B and A.
Q1

How are the areas related to one another? How are they related to the
areas of the squares?

24 • Chapter 2: Proofs and Demonstrations

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Ann Condit’s Proof (continued)
Prove
You may have discovered in the investigation that the sum of the areas of
the smaller triangles is equal to the area of triangle DBK. If you can prove
this is true, and if you can relate these areas to the areas of the squares,
then you will have proved the Pythagorean theorem. Here are some steps
for the proof.
Q2

Triangles DCG, DCF, and DBK share a side length. What sides in these
triangles are equal? Why?

Q3

Segments PF and PG are altitudes of triangles DCF and DCG
respectively. (In Loomis’s book, this is assumed. You can try to prove
it. Hint: Start by showing ∆FGC ≅ ∆BAC. Then show that m∠FPC = 90°
by showing that ∠CBA ≅ ∠GFC and ∠FCP ≅ ∠CAB.) The sum of these
altitudes, FG, is equal to BK. Why?

Q4

Show that area DCG + area DCF = area DBK.

Q5

How does area DCF compare to area CFEB? Why? How do areas DCG
and DBK compare to the areas of the other two squares?

Q6

Now you’re on your own. Combine your answers to the questions
above to write a proof of the Pythagorean theorem.

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Chapter 2: Proofs and Demonstrations • 25

Leonardo da Vinci’s Proof

Name(s):

Leonardo da Vinci (1452–1519) was a great Italian painter, engineer, and
inventor during the Renaissance. He is famous for, among other things,
painting the Mona Lisa. He is also credited with the following proof of the
Pythagorean theorem.
Sketch and Investigate
In this figure, you
don’t have to
construct the square
on the hypotenuse.

1. Construct a right triangle and
squares on the legs.

B
c

2. Connect corners of the squares to
construct a second right triangle
congruent to the original.

a
A
C

3. Construct a segment through the
center of this figure, connecting
far corners of the squares and
passing through C.

b
D

4. Construct the midpoint, D, of this
segment.
The Action Button
submenu is in the
Edit menu.

Hide Reflection

5. This segment divides the figure
into mirror image halves. Select
all the segments and points on one side of the center line and create a
Hide/Show action button. Change its label to read Hide Reflection.
6. Press Hide Reflection. You should now see half the figure.
B
c
a
A
C

b
D

Show Reflection

26 • Chapter 2: Proofs and Demonstrations

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Leonardo da Vinci’s Proof (continued)
7. Mark D as center and rotate the entire figure (not the action button)
180° around D.
8. Select all the objects making up the rotated half of this figure and
create a Hide/Show action button. Relabel the button to read Hide
Rotation, but don’t hide the rotated half yet.
B
c
a
A
C

b
D

Show Reflection
Hide Rotation

9. Construct xA´B and xB´A. Do you
see c squared?

B
c

10. Construct the polygon interior of
BA´B´A and of the two triangles
adjacent to it.
11. Select xA´B, xB´A, and the three
polygon interiors and create a
Hide/Show action button. Name
this Hide c squared.
Going through this construction
may give you a good idea of how
da Vinci’s proof goes.
12. Press each of the Hide buttons,
then play through the buttons in
this sequence: Show Reflection,
Show Rotation, Hide Reflection,
Show c squared.

a
A
C

b
D

A'

B'

Show Reflection
Hide Rotation
Hide c squared

You should see the transformation from two right triangles with squares
on the legs into two identical right triangles with a square on their
hypotenuses.
Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Chapter 2: Proofs and Demonstrations • 27

Leonardo da Vinci’s Proof (continued)
Q1

Explain to a classmate or make a presentation to the class to explain
da Vinci’s proof of the Pythagorean theorem.

Q2

Da Vinci’s is another of those elegant proofs where the figure tells
pretty much the whole story. Write a paragraph that explains why the
two hexagons have equal areas and how these equal hexagons prove
the Pythagorean theorem.

28 • Chapter 2: Proofs and Demonstrations

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Presidential Pythagoras

Name(s):

James A. Garfield discovered a proof of the Pythagorean theorem in 1876,
a few years before he became president of the United States. An interest in
mathematics may not have been a prerequisite for the presidency, but it
must have been common at the time. One of Garfield’s predecessors,
Abraham Lincoln, credited Euclid’s Elements as being one of the books
most influential to his career as a lawyer and politician, saying he learned
from it how to think logically. Garfield’s Pythagorean theorem proof is
illustrated with a relatively simple figure: a trapezoid.
Sketch and Investigate
1. Construct a right triangle ABC and label
it as shown.

D

A'

2. Mark point B as center and rotate side c
and point A by 90°.
3. Connect points A and A´ and construct a
line through point A´, parallel to side b.
4. Use the Ray tool to extend side CB and
construct the point of intersection, D, of
this ray and the line through A´.
5. Hide the ray and the line and replace
them with segments BD and DA´.

B
c
a
C

b

A

6. Construct polygon interiors for the three
right triangles.

The point here is to
use only side lengths
and Calculate
from the Measure
menu to calculate
areas. Don’t actually
measure areas until
you’re ready to
confirm your
calculations.

Q1

What kind of figure is quadrilateral ACDA´? How do you know?
Drag points A, B, and C. Does the figure remain this type of special
quadrilateral?

Q2

What can you say about the triangles into which ACDA´ is divided?
Do they maintain these properties when you drag different parts of
your sketch?

7. Measure sides a, b, and c. Now use just these measures to calculate the
areas of the three triangles and their sum.
8. Now use the area formula for a trapezoid to calculate the area of
ACDA´ using just the side lengths. (What’s the height of trapezoid
ACDA´?) Construct the polygon interior of the entire figure and
confirm your calculations were done correctly.

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Chapter 2: Proofs and Demonstrations • 29

Presidential Pythagoras (continued)
Prove
In steps 7 and 8, you calculated the area of the trapezoid in two different
ways. Garfield used these two different ways of finding the area to prove
the Pythagorean theorem. Can you do it too? Write two different
expressions for the same area in terms of a, b, and c. Set these expressions
equal and do the necessary algebra to arrive at the Pythagorean theorem.

Explore More
Look at the figure in the activity The Tilted Square Proof. How is
Garfield’s figure related to this one? See if you can transform (that’s a
hint) Garfield’s figure into the Tilted Square figure. Compare the algebra
involved in the two proofs and write a paragraph about how the proofs
are related.

30 • Chapter 2: Proofs and Demonstrations

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Perigal’s Proof

Name(s):

Many proofs of the Pythagorean theorem have ancient origins, but were
rediscovered later by people unfamiliar with the older sources. This proof
was “discovered” by mathematician Henry Perigal in 1873, but was
probably known to the Arabian mathematician Tâbit ibn Qorra a
thousand years before.
Sketch and Investigate
1. Construct a square CADE.

E

b

2. Construct a smaller square
adjacent to the original square
so that the two squares share a
vertex and a second vertex is
attached to side AD (square
AGFB in the figure). Label the
sides of these squares b and a.
Select points A and
B in that order and
choose Mark
Vector from the
Transform menu.
Then choose
Translate from the
Transform menu.

3. Mark AB as a vector and
translate point C by this vector.

D

G

F
a

C

C'

A

B

4. Construct sEC´ and sC´F.
5. Construct polygon interiors of the triangles.
Q1

You started with two adjacent squares, and within this figure you
constructed two right triangles. How are these triangles related to one
another? What are the lengths of their legs? Call the length of one of
the hypotenuses c. What’s the length of the other triangle’s
hypotenuse? Drag point G to confirm these relationships hold for
other squares.

6. Use your Translator tool to
translate ∆ECC´ from point C to
point G and to translate ∆C´BF
from point B to point D.
7. Mark point E as center and
rotate point C´ about it 90° to
make square EC´FC´´.
Q2

C''

E
b

Explain how this investigation
demonstrates the Pythagorean
theorem.

G

F
a

C

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

D

C'

A

B

Chapter 2: Proofs and Demonstrations • 31

Perigal’s Proof (continued)
Your answer to question Q2 might serve as a good proof of the
Pythagorean theorem. But it still may seem a surprise that this simple
transformation from a squared and b squared into c squared always
works. Here are some questions meant to guide you toward more insights
into why this proof works.
Q3

When you translate ∆ECC´ into ∆C´´GF, how do you know ∠ECC´ will
fit into ∠C´´GF? How do you know side CC´ fits into side GF?

Q4

Side CE, with length b, was translated to side GD. This new side
extended above the figure by a length DC´´. How do you know this
length is a? Hint: What’s the length GD?

Q5

How do you know quadrilateral EC´FC´´ is a square?

Q6

Combine your answers to these questions into a more complete
paragraph proof of the Pythagorean theorem.

32 • Chapter 2: Proofs and Demonstrations

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Euclid’s Proof, a.k.a.
Pythagoras’ Pants

Russian students
less reverent than
Arab scholars have
called the figure in
Euclid’s proof
Pythagoras’ pants.

Name(s):

The Pythagorean theorem is one of the important milestones in Euclid’s
Elements. This work, written around 300 B.C.E., has had a tremendous
influence on mathematics because of the systematic way in which it
presents geometry propositions logically derived from one another. Euclid
arrives at the Pythagorean theorem and its converse as the 47th and 48th
(and final) propositions of Book 1 (out of 13). It’s thanks to Arab scholars
and Moorish scholars of northern Africa and southern Spain that much of
ancient Greek mathematics survived. The figure on page 7 is from a
manuscript of Tâbit ibn Qorra’s translation of Euclid. Arab scholars
referred to the figure as “the figure of the bride.”
Sketch and Investigate
1. Construct a right triangle and squares on the sides, as shown.
2. Construct a line through the right angle vertex, perpendicular to the
hypotenuse.
3. Construct sCF and sBE.
D

E
C

B

A

G

F

Euclid’s proof is pretty quick, if you can establish the relationships
between the triangles and the squares. But those relationships can be
subtle in the complex diagram. A little further construction can add a
dynamic element to your sketch that will give you more insight into how
the proof works.

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Chapter 2: Proofs and Demonstrations • 33

Euclid’s Proof (continued)
4. Construct a point K on side
BC. Construct the polygon
interior EAK. You should be
able to drag point K back and
forth along sBC.
Q1

Q2

When point K coincides with
point C, what can you say
about the relationship
between the shaded triangle
and square ACDE?

D

E
C
K

B

Drag point K toward point B.
G
Does the area of the triangle
change? Explain why this is
so. Hint: If you consider AE
as the length of the base,
what’s the height of ∆EAK? Does that change?

J

H

A

F

5. Drag point K until it coincides with point B. Now mark A as center,
select the triangle interior, and rotate it by 90°.
Q3

What new triangle interior do you get? How are triangles EAB and
CAF related?

Q4

In questions Q1 and Q2, you established a relationship between ∆EAB
and square EACD. How are ∆CAF and rectangle JAFH related?

Prove
The investigation gives you a start on Euclid’s proof. First, show
∆EAB ≅ ∆CAF. Then supply the necessary steps to show the area
of square ACDE is equal to the area of rectangle AJHF. A similar
argument will prove the area of the small square is equal to the area
of rectangle BGHJ.

34 • Chapter 2: Proofs and Demonstrations

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

A Four-Piece Dissection

Name(s):

Until now, all the dissection demonstrations in this book involved at least
five pieces. In this activity, you’ll manipulate a pre-made sketch in which
the squares on the legs are divided into four pieces that can be arranged to
fit into the square on the hypotenuse.
Investigate
1. Open the sketch 4-Piece
Pythagoras.gsp.
In the sketch, the square on side b
is divided into three pieces which,
combined with the square on side
a, correspond to four pieces in the
square on side c. Here’s how the
dissection is done:
• The square on side b is
divided in half. Region 1 is
placed in one corner of the
square on side c.

3
A
2
3
1

b

c

1

a

2
C

B

• There will be some area left
over on the side of the square on side c. This region, labeled region 2,
is marked off in the other half of the square on side b.
• The remaining region in the square on side b, region 3, will fit with the
square on side a to fill the square on side c.
Q1

To confirm this dissection works, we need to manipulate the triangle
into different sizes and shapes. Start by dragging point C. Does the
dissection hold up? Now drag A or B. What happened?

You may have been fooled for a minute into thinking this dissection
works for any right triangle. But, in fact, there is no dissection proof of the
Pythagorean theorem with fewer than five pieces. Still, it seemed to hold
up when you dragged C.
Q2

What can you say about the triangles you formed dragging point C?

Q3

You should have observed that the dissection pieces don’t stay
identical when you drag point A or point B. Undo until the pieces
are identical again. What’s special about this triangle? Make some
measurements and calculations to see what you can discover. Can
you explain why this works for this type of triangle and no other?

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Chapter 2: Proofs and Demonstrations • 35

Tessellating Pythagoras

Name(s):

To tessellate or tile the plane means to cover it with closed shapes that fit
together without overlap or gaps. Both simple and highly decorative
tessellations are common in floor and wall designs, especially in Islamic
art. In this activity, you’ll create a decorative pattern that demonstrates the
Pythagorean theorem.
Sketch and Investigate
1. Construct a square.
2. Construct a smaller square adjacent to it, using a
point on one side of the original square and one
of its vertices.
3. Mark AB as vector and translate the entire
figure by this vector. Repeat this translation two
or three times.

B
A

B
A
Select the entire
figure by dragging a
selection marquee
around it or by
choosing Select
All from the
Edit menu.

4. Mark CD as
vector and
translate the
entire figure by
this vector two or
three times.

Make sure to drag
the points you
started with to
manipulate your
tessellation. This
dynamic tessellation
can be quite
dazzling. Once
you’ve tessellated
with c squared,
shade or color every
other c squared
to get a checkerboard effect.

You’ve begun to
tessellate with two
squares of different
sizes. Note that you
could continue this
tessellation forever to
fill an infinite plane.
To demonstrate the
Pythagorean theorem
with the tessellation,
think of these squares
as a squared and b
squared. It’s possible

36 • Chapter 2: Proofs and Demonstrations

D
B
A

C

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Tessellating Pythagoras (continued)
to also tessellate with c squared, where a, b, and c are legs of a right
triangle. It’s left to you to find points you can use for a square with
side c—you actually have a number of choices. Find four points you can
use to construct the interior of one square with side c and tessellate with it.
Prove
Write a paragraph explaining how this tessellation proves the
Pythagorean theorem. Here are some questions to help guide your proof.
Q1

Why is it possible to tessellate with two squares of different sizes?

Q2

How do you know that the squares you constructed with side c are
really squares? (Assume, as obvious, that it’s possible to tessellate by
translating a single square.)

Q3

For every a squared and b squared pair in the two-square tessellation,
how many c squareds are there?

Explore More
Experiment with different polygon interior shades,
colors, and arrangements to make your tessellation
look nice. Make an Animation action button that
moves point B on your original square
bidirectionally along the side it’s on.

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

B
A

Chapter 2: Proofs and Demonstrations • 37

A Dynamic Proof

Name(s):

Unlike dissection proofs, in which you move pieces to different locations
in your figure without changing them, in this construction you’ll
transform your squares, without changing their areas, to create congruent
shapes. To do this, you’ll actually construct parallelograms on the sides of
your right triangle.
Sketch and Investigate
1. Construct a right triangle and
squares on the sides.
2. Hide the far side of the square on
side a and replace it with a line.
In steps 3
through 5, you’ll
construct a
parallelogram
on side a.

B

3. Construct sCF, where F is a point on
the line.

C

4. Construct a line through B, parallel
to sCF.

c

a

G

b

A

F

5. Construct the interior of
parallelogram BCFG and hide
the lines.
This transformation
is called a shear. A
shear translates
every point in a
figure in a direction
parallel to a given
line by a distance
proportional to the
point’s distance from
the line. Shearing a
figure preserves
its area.

Drag point F to experiment with the parallelogram’s behavior. The key to
this demonstration is that the parallelogram’s area is always the same as
the square whose side it shares. Can you see why?
6. Construct a parallelogram on side b by the same method you used in
steps 2 through 5 above, starting with segment CH so that H is a point
that can be dragged.

B

c

a

G

b
A

C
F

H
38 • Chapter 2: Proofs and Demonstrations

J

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

A Dynamic Proof (continued)
You should now have parallelograms on sides a and b that can be dragged
back and forth by one vertex without changing their areas.
7. Construct a line through
point C, perpendicular to
side c.

L

8. Construct sBK and sAK,
where K is a point on
this line.
9. Mark BL as a vector and
translate sBK, point K, and
sAK by this marked vector.
Note that this
concave hexagon
consists of two
parallelograms,
BLK´K and AKK´M.

M

K

B

c

a

G

K'

C

b
A

10. Construct the polygon
interior BLK´MAK.
F

11. Drag point K to make sure
polygon BLK´MAK behaves
correctly: Point K´ should
move when you drag K so
that BLK´K and AKK´M
remain parallelograms.

H

J

If everything is working properly, you’re ready to manipulate your sketch
to give a dynamic proof of the Pythagorean theorem.
In your sketch, you
may need to drag a
different point,
depending on how
you constructed the
parallelogram.

12. Drag point F to shear the parallelogram back and forth. Note that the
parallelogram can fill the square. Note too that the parallelogram’s
area doesn’t change. (You can measure to confirm this.) Finish the
shear by dragging F to lie on the line through C and K.
Q1

Explain why the area of the parallelogram doesn’t change.

13. Shear the parallelogram on the other leg of the triangle. Note that it
too fills its square and that its area doesn’t change. Drag until point H
is on the line.
14. Drag point K to shear the parallelograms in the square on c. Note that
these parallelograms can fill the square and that the area of the
polygon (the sum of the areas of the parallelograms) doesn’t change.
Drag until point K coincides with point C.
Q2

The figures on the next page show the sequence you might have
performed. How does this demonstrate the Pythagorean theorem?

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Chapter 2: Proofs and Demonstrations • 39

A Dynamic Proof (continued)

Stage 1

Stage 2

Stage 3

Stage 4

Prove
The dynamic “proof” you perform by manipulating your sketch to
transform the squares into two congruent figures may seem proof enough
of the Pythagorean theorem. People might consider the details of the proof
superfluous compared to the convincing power of the dynamic
demonstration. Yet if you were showing this proof to someone, you’d
probably want to explain what’s going on and why your sketch proves the
theorem. And to make it a complete, or rigorous, proof, you’d want to
supply the details. Start with a figure like that in stage 4 above. Add labels
and write a complete proof. That is, show that each parallelogram is equal
in area to a square (or part of a square) and show that the two pairs of
parallelograms are congruent.

40 • Chapter 2: Proofs and Demonstrations

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

A Dynamic Proof (continued)
Explore More
Write a paragraph about any similarities you see between this proof and
Euclid’s proof.
A Presentation Sketch
This sketch lends itself nicely to being made into a presentation sketch—a
sketch whose actions can be driven by double-clicking action buttons. In
the figure below, point C´ is the translated image of point C by vector LB.
Make the action buttons indicated. The Shear button is a Presentation
button that sequences the first three Move buttons. The Return to Squares
button presents the second set of three Move buttons simultaneously.
Once the Presentation buttons are made, hide the separate Move buttons.

L

B
K
G

Move F -> C'

N

Move H -> C'

C

J

Move K -> C

Move F -> J

F

Move H -> K
Move K -> N
C'

H
Shear

K
Return to Squares

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Chapter 2: Proofs and Demonstrations • 41

A Coordinate Proof

Name(s):

This unique proof involves putting a right triangle in the coordinate plane
and using a property of slopes of perpendicular lines.
Sketch and Investigate
1. Choose Define
Coordinate System
from the Graph menu
and turn off the grid.
Show the label of the
origin point. Hide the
unit point (1, 0), taking
care not to move it.

B

2
c

D

2. Construct a point B
anywhere in the first
quadrant.

A

a
b

C

E

-2

3. Construct a line
through B perpendicular to the x-axis.
4. Construct point C where this line intersects the axis.
5. Hide the line and construct segments to make ∆ABC. Label the
horizontal leg b, the vertical leg a, and the hypotenuse c.
6. Measure a, b, and c. Also measure the coordinates of point B.
Q1

Describe how the coordinates of point B are related to the side lengths
of the triangle.

7. Construct circle AB.
8. Construct points D and E, the two points of intersection of this circle
with the x-axis.
9. Measure ∠DBE.
Q2

What do you notice about this angle measure? Does it change when
you drag point B? Why or why not?

10. Measure the coordinates of both D and E.
Q3

Describe how these coordinates are related to a side length in the
triangle and why.

42 • Chapter 2: Proofs and Demonstrations

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

A Coordinate Proof (continued)
11. Construct segments BD and BE.
12. Measure these segments’ slopes.
13. Perform a calculation on one of the slopes that gives a value equal to
the other slope.
Q4

Use the definition of slope to write the slopes of DB and EB in terms of
a, b, and c.

Q5

Use what you know about the slopes of perpendicular lines to write
an equation from these slopes. Simplify the equation to get the
Pythagorean theorem!

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Chapter 2: Proofs and Demonstrations • 43

The Similar Triangle Proof

Name(s):

The Pythagorean theorem proof most commonly found in geometry books
is based on similar triangles. In this activity, though, you’ll do more than
simply prove the theorem. You’ll puzzle out how similar triangles can
be transformed into one another, and in the process you’ll discover a
surprising generalization of the Pythagorean theorem.
Sketch and Investigate
1. Construct a right triangle.
2. Construct a line through the right
angle vertex C perpendicular to the
hypotenuse.

C

3. Construct a segment CD, where D is
the intersection of the perpendicular
line and the hypotenuse. Then hide
the line.

B

A

D

4. Construct the polygon interior of CBD.
Negative numbers
give clockwise
rotations.

5. Mark point D as center, select the interior of triangle BDC, and rotate
it by −90°.
You should have a copy of ∆BDC positioned in the right angle corner of
∆CDA. This rotated copy may make it easier to see a relationship between
∆BDC and ∆CDA.
C

C

B
Mark Ratio and
Dilate are both
commands found in
the Transform menu.
Select two segments
and choose Mark
Ratio. Then select
the figure you want
to dilate and choose
Dilate. Make sure
By Marked Ratio
is chosen in the
Dilate dialog box.

D

A

B

D

A

6. Now you want to expand the rotated triangle to fill triangle CDA. To
do this, select two segments whose ratio defines a scale factor that will
enlarge the small triangle to fill ∆CDA. Mark this ratio, and mark
point D as a center for dilation. Dilate the rotated triangle by this
marked ratio.
Q1

How are ∆BDC and ∆CDA related? Explain.

44 • Chapter 2: Proofs and Demonstrations

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

The Similar Triangle Proof (continued)
Q2

Compare these two triangles to the whole triangle, BCA. Complete the
following similarity statements:
∆

˜∆

˜∆

7. Measure sides a and b and areas
BDC and CDA. Calculate a2 and b2,
and compare these calculations to
the areas.
Q3

Q4

Write an equation relating a2, b2, and
the two area measurements.

C

a

B

b

D

c

A

Consider the following statements:
area ∆BDC = ka2 and area ∆CDA = kb2.
Explain what these statements mean and why they are true based on
your observations in step 6 and question Q3.

Q5

Can a similar statement be made relating the area of ∆BCA to side c?
Measure and do calculations to confirm.

The investigation above highlights a very important feature of similar
figures. By definition, corresponding lengths in similar figures are
proportional. But the ratio of corresponding areas in similar figures is
equal to the square of the ratio of corresponding lengths. So the areas of the
similar triangles in your construction are proportional to the squares of
their corresponding hypotenuses. The hypotenuses of the three triangles
are a, b, and c. The area of the triangle with hypotenuse a can be written
ka2, where k is some constant of proportionality.
Q6

Clearly, area ∆BDC + area ∆CDA = area BCA. Write an equation
relating a, b, c, and a proportionality constant, k. You’ve written a
one-line proof of the Pythagorean theorem!

Explore More
The proof commonly found in geometry uses a proportion of
corresponding sides in the similar triangles. Let BD in the figure above
equal x. Write a proportion involving a, b, c, and x and simplify it to get
the Pythagorean theorem.
.

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Chapter 2: Proofs and Demonstrations • 45

Special Cases and Variations

Renaissance artist Leonardo da Vinci proved the Pythagorean theorem and
may have also known of this corollary: “The sum of the Mona Lisas on the
legs of a right triangle is equal to the Mona Lisa on the hypotenuse.”

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Chapter 3: Special Cases and Variations • 47

The Isosceles
Right Triangle

Name(s):

The isosceles right triangle played an important role in another great
achievement of the Pythagoreans: the discovery of irrational numbers. An
irrational number is a number that cannot be written as a simple ratio, p/q,
where p and q are integers. This discovery was something of a blow to
Pythagorean philosophy, which was based on the belief that whole
numbers and ratios of whole numbers described the essence of the
universe.
In this activity, you’ll discover what was probably the first-discovered
irrational number—the length of the diagonal of a unit square.
Sketch and Investigate
1. Choose Preferences from the Edit menu. Set
the Distance Unit to inches and the Distance
Precision to hundred-thousandths.
2. Construct a point A.
By default, Polar
should be chosen in
the Translate dialog
box. Point A´ is the
translated image.

3. Use the Translate command from the
Transform menu to translate it by 1 inch
with a 0° angle.

A

A'

4. Use these two points to construct a square.
5. Construct a diagonal of the square.
Q1

Explain why the diagonal divides your square into two isosceles
right triangles.

Q2

Use the Pythagorean theorem to show why the length of this diagonal
is

2 inches. Now measure the diagonal in your sketch to get an

approximation of 2 to the nearest hundred-thousandth. If you have
a calculator handy, use the square root key to confirm the precision of
Sketchpad’s measurement. Copy and complete this statement on your
paper.

2 ≈
6. Use Sketchpad’s calculator to multiply the length of the diagonal by
100,000. This allows you to see more decimal places in
Q3

2.

Rational numbers have the characteristic that their decimal parts (if
they have one) always either end or repeat. At least at the precision
you see, does it appear that the decimal part of

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

2 ends or repeats?

Chapter 3: Special Cases and Variations • 49

The Isosceles Right Triangle (continued)
A geometric consequence of irrational numbers is that they give lengths
that are incommensurable. A simplified definition of incommensurable
lengths is that they are lengths that cannot be measured exactly with the
same ruler. This may have been what troubled the Pythagoreans the most.
To understand what
this means, suppose
you started with a
square whose side
measured 1 meter.
You could measure
its diagonal with a
ruler divided into
tenths, hundredths,
thousandths, and so
on, but because the

2
1.4

1

1.5

2

2

1.41

1.42

2

1.414
1.415
decimal part of 2
never ends, you’d
never get a mark
that lined up perfectly with the diagonal length.

Some stories
(probably untrue)
have it that the
Pythagoreans were
so upset by the
discovery of
irrational numbers
that they banished or
killed the discoverer.

Can you see why that might have bugged the Pythagoreans, who believed
numbers described everything? What’s perhaps most mind-boggling is
that a number that can’t be measured as a fraction of a whole number
unit can be so easily constructed! It’s just the diagonal of a unit square.
In fact, you can construct the mark for 2 on your ruler if you want. You
just can’t get to it by dividing the units on your ruler into fractions.
Prove
The Pythagoreans probably proved that 2 was irrational using an
indirect proof like the one outlined below. See if you can follow the
outline and write a proof on your own paper, filling in where the outline
leaves question marks.
Assume 2 is rational and can be written as the ratio of two whole
numbers, reduced to its lowest terms:

2 =

p
q

(Any fraction can be reduced to its lowest terms, and the assumption that
p/q is reduced means that p and q have no common divisors.)
Square both sides of the equation, then multiply by q2 to get
p2
= (?)
q2
50 • Chapter 3: Special Cases and Variations

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

The Isosceles Right Triangle (continued)
How do you know
that if a square
number is even its
square root is also
even? Think about
the prime factorization of an even,
square number.

Multiply by q2 to get
p2 = (?)
So p2 is an even number, and thus so is p. Substitute 2r for p. That gives
(2r)2 = 2q2, or
(?)r2 = 2q2
q2 = (?)
So q is also an even number, leading to a contradiction to our assumption
that (?).
Explore More
1. Hide half the square you made in the first part of this activity so
you have an isosceles right triangle with 1-inch legs. Construct
squares on the sides of this triangle. What do you know about the area
of the square on the hypotenuse? Does this give you an idea of where
the term square root comes from?
2. The figure you constructed in the first part of this activity isn’t very
dynamic. You can’t change its size or position it any way but
horizontally. Start in a new sketch by constructing any line segment,
making it a leg of a more general isosceles right triangle. Measure the
hypotenuse and a leg and calculate the ratio of these lengths. How are
the lengths of the hypotenuse and a leg of any isosceles right triangle
related? Describe a shortcut for finding the length of the hypotenuse
of an isosceles right triangle if you know the length of a leg.

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Chapter 3: Special Cases and Variations • 51

The 30°-60°-90° Triangle

Name(s):

The 30°-60°-90° triangle, half an equilateral triangle, has properties that
make it easy to find the other side lengths if you know one side length.
This construction
takes advantage of
the fact that an
equilateral triangle is
also equiangular.
Can you find other
methods for
constructing an
equilateral triangle?

Sketch and Investigate
1. Construct segment AB.

B'

2. Mark point A as center and rotate the
segment and point B by 60°.
3. Connect points B´ and B.
4. Construct point C, the midpoint of sAB,
and construct xB´C.
Q1

You constructed segment B´C to be a
median, but what else can you say
about that segment? What are the
measures of angles ACB and AB´C?

A

C

5. Hide the right half of your triangle and supply the
new segments necessary to make 30°-60°-90°
triangle ACB´.

B

B'

6. Measure lengths and ratios, if necessary, to look
for relationships between AC and AB´ and
between AC and B´C.
Q2

Q3

Let AC = a and write an expression for AB´ in
terms of a. Use the Pythagorean theorem to write
an expression for CB´ in terms of a.

Use Sketchpad to calculate
hundred-thousandth:

A

C

3 to the nearest

3 ≈
Explore More
1. Prove that 3 is irrational by indirect proof. First, suppose it is
rational and can be written as p/q, where p and q are integers with no
common divisors. Then show that if p/q =
a common divisor of 3.

3 , then p and q must have

2. Construct all three medians in your equilateral triangle. What kinds of
triangles are formed? Call the shortest segment in your figure x, then
express as many lengths as you can in terms of x.
52 • Chapter 3: Special Cases and Variations

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

The Square Root Spiral

Name(s):

Irrational numbers like 2 and 3 correspond to points on a ruler, but
you could never find those points precisely by dividing your ruler into
fractional parts. Interestingly, you can construct square roots with
compass and straightedge (or with Sketchpad). In this activity, you’ll
construct a square root spiral and a table of approximate square roots.
Sketch and Investigate
The first part of
the spiral is a right
triangle with one
1-inch leg. Make
sure to choose
inches for your
Distance Unit under
Preferences
in the Edit menu.

1. Construct a segment AB.
2. Translate point B by 1 inch
and 0° to construct point B´.

A

3. Construct circle BB´.
4. Construct a line perpendicular
to sAB. Drag, if necessary, so
that the line does not appear to
go through point B´.

C

B

B'

5. Construct point C at the
intersection of the circle and
the perpendicular line.
6. Hide the circle, point B´, and
the line.
7. Construct sBC and sAC.
8. Measure sAC.
So far you have a right triangle in which one leg can be any length and the
other is always 1 inch long. Next, you’ll iterate this construction to create a
spiral of right triangles with 1-inch legs.

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Chapter 3: Special Cases and Variations • 53

The Square Root Spiral (continued)
9. Select points A and B and choose
Iterate from the Transform menu.
10. Map point A to itself and point B
to point C, then click Iterate.
11. With the iterated image selected,
use the + and – keys on the
keyboard to increase or decrease
the number of iterations (and thus
the number of triangles and
entries in the table).
12. Drag point A or point B and
observe the effect on the spiral.
This changeable spiral is fun to
manipulate, but we want to look at a
specific spiral—one in which the first
spiral arm has a length of 1 inch.

D

13. Construct a point D anywhere
in the sketch and translate it by
1 inch and –90° to make a point D´
directly below it.

D'

C

14. Merge point A to point D, then merge point B to point D´.
Q1

Study the table generated by the iteration. The second entry in the
table (n = 1), corresponds to the second spiral arm—the hypotenuse in
the original construction. What is this length in radical form?

Q2

What does the third entry in the table (n = 2) represent? Use the
Pythagorean theorem to find this number in radical form.

Q3

Make a table on your paper of square roots and their decimal
approximations for the numbers 1–16.

Q4

Explain why this construction generates square roots of consecutive
whole numbers.

54 • Chapter 3: Special Cases and Variations

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

A Pythagorean Tree

Name(s):

With Sketchpad’s iteration capabilities, you can construct part of the
Pythagorean theorem figure (a square with a right triangle on one side),
and then iterate the construction as many times as you like to create a
fractal-like Pythagorean tree. The tree isn’t likely to provide any new
insight into the theorem, but it’s fun to look at and play with!
Sketch and Investigate
1. Construct a square ABCD that’s controlled by
the two bottom points A and B.
2. Construct the midpoint E of the top side CD and
construct circle EC.

F
E
D

C

A

B

3. Select, in order, point C, point D, and the circle.
Choose Construct|Arc on Circle.
4. Hide the circle and construct point F on the arc.
5. Construct sides CF and DF.
Steps 6 and 7 will
add some color
“special effects” to
your sketch when
you’re done.

6. Measure one side of the square.
7. Select this measurement and the square interior and choose
Display|Color|Parametric. Enter a range from 0 to about what the
current side length is.
8. Select points A
and B and choose
Transform|Iterate.
Map points A and B
to points A and F. In
the Structure pop-up
menu, choose Add
New Map and map
points A and B to
points F and B. Also in
the Structure popup menu, uncheck
Tabulate Iterated
Values. Click Iterate.

You can add
branches by
selecting any part of
the iterated image
and pressing + or –
on your keyboard.

You should now have
squares and right triangles growing off the legs of your original right
triangle. The colors of the squares are determined by their side lengths.
Drag point F to wave the tree’s branches and watch these colors change.
Q1

What’s the relationship between any triangle in the figure and the two
triangles in the next iteration?

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Chapter 3: Special Cases and Variations • 55

Unsquare Pythagoras

Name(s):

In the activity The Similar Triangle Proof, you arrived at the Pythagorean
theorem by way of an even more general statement, stated algebraically as
ka2 + kb2 = kc2, where k is a constant. Whereas the algebraic terms a2, b2,
and c2 can represent the areas of squares with side lengths a, b, and c, the
terms ka2, kb2, and kc2 can represent the areas of any three similar figures
whose areas are proportional to a2, b2, and c2. In this activity, you’ll use
transformations to construct similar figures on the sides of your right
triangle.
Sketch and Investigate
1. Construct an arbitrary
triangle ABC (not
constrained to be a right
triangle) with longest side c
and shortest side a.

C'

2. Construct an arbitrary
pentagon (or any polygon)
and its interior on side BC.

C

4. Mark angle C´CA as an
angle for rotation and mark
point C as center. Rotate
the translated interior.

b

a

3. Translate the interior and
point C by vector BC.

c

A

B

You now have a pentagon on side b identical to the pentagon on side a.
Now you just need to stretch it to the right size.
5. Select side b and side a and
choose Mark Segment Ratio
from the Transform menu.
C

6. Select the polygon interior on
side b and choose Dilate from the
Transform menu.
7. Hide the two intermediate
interiors: the translated one and
the smaller interior on side b.

a

b
c

B

A

Rather than going through that whole
process again to get a polygon on
side c, you can make a tool to do it
quickly.

56 • Chapter 3: Special Cases and Variations

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Unsquare Pythagoras (continued)
8. Select, in order, the polygon on side a, the polygon on side b, point B,
point C, and point A. Choose Create Tool from the Custom Tools
menu. Name the tool, if you wish.
9. Choose the tool and use it by clicking on, in order, the polygon on
side b, point C, point A, and point B. If your tool works correctly, it
should construct a similar polygon on side c as you drag from point A
to point B.
10. Now to make the triangle a right triangle, construct the midpoint D of
side c and construct circle DA. Merge point C to the circle.
11. Measure the areas of the polygons, and calculate the sum of the areas
of the polygons on sides a and b.
Q1

What do you observe about the sum of the areas? If you can, explain
why this is so. If you’re unsure, continue to question Q2.

Q2

Calculate the ratio of each area to its corresponding side length a, b, or
c. You’ll notice that these ratios are not equal. Experiment with editing
the calculations until you get equal ratios.

Q3

The ratio you found in question Q2 is the proportionality constant k in
the generalized Pythagorean equation ka2 + kb2 = kc2. Explain what this
equation means in relation to the polygons on the sides of the right
triangle.

Q4

The proportionality constant k will be the same for any three similar
shapes on the sides of a right triangle. But a different set of three
shapes would have a different k value. What would the value of k be
for semicircles on the sides of a right triangle?

Q5

What would the value of k be for equilateral triangles on the sides of a
right triangle?

Q6

What is the value of k for squares on the sides of a right triangle?

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Chapter 3: Special Cases and Variations • 57

Pappus’ Theorem

Name(s):

Pappus was a Greek geometer who lived in the third century C.E.,
contributing original geometry propositions at a time when the golden age
of the Greek mathematicians Pythagoras, Euclid, Archimedes, and
Appolonius was beginning to fade. His theorem describes a relationship
that holds for areas on the sides of any triangle, not just right triangles. The
Pythagorean theorem, in fact, is just a special case of Pappus’ theorem.
Sketch and Investigate
1. Construct any triangle ABC.
2. Construct parallelograms of
any size on two sides.

D

3. Construct rays to extend the
sides of the parallelograms
and construct their point of
intersection, D, as shown.
4. Construct ray DB and point E
where the ray intersects sAC.

B

A

E

C

To translate by a
vector you first need
to select the vector
endpoints and
choose Mark
Vector from the
Transform menu.
Then choose
Translate.

5. Translate point E by
vector DB.

If you did the activity
A Dynamic Proof,
this figure should
seem familiar.

8. Measure the areas of the parallelograms and look for a relationship
among these areas.

6. Construct a line through E´,
parallel to sAC.

G

E'

F

7. Construct parallelogram
ACFG, where sFG is on the line
through point E´. Also construct the interior.

Q1

Q2

How are the parallelograms related? Drag different points to change
your triangle and the first two parallelograms you constructed. Does
the relationship hold for any triangle? For any parallelograms
constructed on two sides?

Drag point F (or point G, if point F doesn’t cooperate) so that sCF and
sAG are parallel to sEE´. Each of the parallelograms on sides AB and BC
are equal to part of parallelogram ACFG when it’s aligned this way.
Which parts are equal and why?

58 • Chapter 3: Special Cases and Variations

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Pappus’ Theorem (continued)
Q3

Manipulate the first two parallelograms so that they approximate
squares. Is the third parallelogram ever a square? Under what
conditions?

Prove
In the Sketch and Investigate
section, you discovered
Pappus’ theorem that the
sum of the areas of two
parallelograms on two sides of
any triangle is equal to the area
of a specially constructed
parallelogram on the third side.
If you were able to answer
question Q2, you discovered
that the third parallelogram
could be manipulated and
then divided into two
parallelograms, each equal in
area to one of the first two
parallelograms. To prove a pair
of parallelograms have equal
area, show that each has area
equal to a third parallelogram.
Points K and L in the diagram
at right should give you a hint.

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

D

L

K

B

A

E

G

C

E'

F

Chapter 3: Special Cases and Variations • 59

The Law of Cosines

Name(s):

The Pythagorean theorem gives you a way to find the length of one side
of a right triangle if you know the lengths of the other two sides.
Trigonometry builds on the Pythagorean theorem with theorems that
enable you to find the measures of all the parts of a triangle given
measures of parts that determine the triangle. With the law of cosines,
you can find the measure of an unknown side of any triangle given the
measures of the other two sides and the angle between them (SAS). In this
activity, your construction will demonstrate the law of cosines for obtuse
triangles. To understand how the figure demonstrates it, you’ll need to
know a little trigonometry.
Sketch and Investigate
1. Construct a square and label the
bottom side c.
Any reference to ∠C
in this activity will
refer to ∠ACB.

2. Construct ∆ABC where C is any
point inside the square that makes
∠C obtuse. Give segments AC and
BC the labels b and a.

G

H

D

F

E

K

3. Construct a square on side b.
Make sure to use
Parallel Line from
the Construct menu.
Don’t just
“eyeball” it.

4. On side a of the triangle, construct
parallelogram CBEF as shown.

C
a

b
A

c

B

5. On the side of this parallelogram
opposite side a, construct another
square.
6. Construct segments GH and HK to complete a second parallelogram,
as shown. Construct the interiors of the parallelograms and squares.
Q1

Drag point C. What do you notice about the two parallelograms (as
long as ∠C is obtuse)? What happens when ∠C becomes a right angle?

Q2

The area of your original square is c2. How does that compare to the
sum of the four areas you constructed (the two squares and two
parallelograms)?

Q3

If you know some trigonometry, write an expression for the area of
one of the parallelograms in terms of a, b, and ∠C.

Q4

Use this expression and what you discovered about the sum of the
areas of the squares and parallelograms to complete the law of
cosines: c2 = (?)

60 • Chapter 3: Special Cases and Variations

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Problems

Even without reading this problem, you can probably guess what it’s
about. This problem is from a work by Yang Hui (1261 C.E.) of China.
See problem 6 on page 64 for a translation.

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

Chapter 4: Problems • 61

Problems

Name(s):

Right triangles can be used to model all kinds of real-world problems.
That’s why the Pythagorean theorem shows up early in the history of so
many cultures’ mathematics.
Use the Pythagorean theorem to solve the following problems. Once
you’ve solved them, model selected problems from the first 12 or make
scale drawings with Sketchpad and take measurements to confirm your
answers.
1. A rope from the top of a sailboat mast is
fastened to the deck 4 meters from the base of
the mast. If the rope is 10 meters long, how
tall is the mast?
10 m

mast

4m

2. I drive at 40 mph from my home to
work. I have to drive 8 blocks south,
then 6 blocks west. (The blocks are all
the same length.) My faithful pet
pigeon decides to follow me to work.
He can fly straight there at 30 mph.
Who will arrive at the office first, me
or my bird?

home

8 bl

office

3. A baseball diamond is a square with 90-ft
sides. The pitcher’s mound lies on the
diagonal between home plate and second
base, 60 ft 6 in. from home plate. Is the
mound in the center of the diamond? If
not, how far off is it?

3rd

2nd

mound

home

Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad
© 2003 Key Curriculum Press

6 bl

90 ft

1st

Chapter 4: Problems • 63

Problems (continued)
4. How long a board should you cut to make a
diagonal brace for a door 2 m high and 1 m wide?

2m

1m

5. This problem is from the Chinese
manuscript Arithmetic in Nine Sections
(1115 C.E.): “There grows in the
middle of a circular pond 10 feet in
diameter a reed which projects 1 foot
out of the water. W