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Exploring Conic Sections with The Geometer's Sketchpad
Exploring Conic Sections with The Geometer's Sketchpad
Daniel Scher
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Few topics connect with as many branches of mathematics as conic sections. In this rich collection of activities, students build their own physical models of conic sections and then construct more flexible models with Sketchpad. And by developing both geometric and analytic proofs, students connect these mathematical realms in ways essential to any secondyear algebra, precalculus, or analytic geometry course.
The sketches referenced in the book can be downloaded from https://sketchpad.keycurriculum.com/KeyModules/index.html.
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Year:
2012
Publisher:
Key Curriculum Press
Language:
english
Pages:
96
ISBN 10:
1604402776
ISBN 13:
9781604402773
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PDF, 4.89 MB
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with DANIEL SCHER Prepress and Printer: Steven Chanan Jennifer Strada Caroline Ayres Jason Luz Joan Saunders Debbie Cogan Diana Jean Parks Ariana Grabec–Dingman Private Collection, Bridgeman Art Library; Paul Eekhof, Masterfile Lightning Source, Inc. Executive Editor: Publisher: Casey FitzSimons Steven Rasmussen Project Editor: Production Editor: Art and Design Coordinator: Art Editor: Copyeditor: Manager, Editorial Production: Production Director: Cover Designer: Cover Photo Credits: Cover art and text include illustrations from Exercitationum Mathematicorum, Liber IV, Organica Conicarum Sectionum in Plano Descriptione by Frans van Schooten (Leiden, 1646). An original copy of this book is housed at Cornell’s Kroch Library. ®Key Curriculum Press is a registered trademark of Key Curriculum Press. ®The Geometer’s Sketchpad and ®Dynamic Geometry are registered trademarks of KCP Technologies. ⇥Sketchpad is a trademark of KCP Technologies. All other brand names and product names are trademarks or registered trademarks of their respective holders. Limited Reproduction Permission © 2012 Key Curriculum Press. All rights reserved. Key Curriculum Press grants the teacher who purchases Exploring Conic Sections with The Geometer’s Sketchpad the right to reproduce activities and example sketches for use with his or her own students. Unauthorized copying of Exploring Conic Sections with The Geometer’s Sketchpad or of the Exploring Conic Sections sketches is a violation of federal law. Key Curriculum Press 1150 65th Street Emeryville, California 94608 5105957000 editorial@keypress.com www.keypress.com 10 9 8 7 6 5 4 3 2 1 ISBN 9781604402773 15 14 13 12 Contents Downloading Sketchpad Documents ........................................................ iv Introduction ....................................................................................................... v Customizing the Activities .............................................................................. vi Acknowledgments .............; ............................................................................... vi Common Commands and Shortcuts ....................................................... viii Chapter 1: Ellipses Chapter Overview................................................................................................ 3 Getting Started: When Is a Circle Not a Circle? ............................................. 5 The PinsandString Construction .................................................................... 6 The Concentric Circles Construction ............................................................... 8 Some Ellipse Relationships .............................................................................. 11 The Folded Circle Construction ...................................................................... 13 The Congruent Triangles Construction ......................................................... 17 The Carpenter’s Construction ......................................................................... 19 Danny’s Ellipse .................................................................................................. 25 Ellipse Projects ................................................................................................... 28 Chapter 2: Parabolas Chapter Overview ............................................................................................. 33 Introducing the Parabola ................................................................................. 34 The Folded Rectangle Construction ............................................................... 38 The Expanding Circle Construction ............................................................... 42 Parabola Projects ............................................................................................... 45 Chapter 3: Hyperbolas Chapter Overview ............................................................................................. 49 Introducing the Hyperbola .............................................................................. 50 The Concentric Circles Construction ............................................................. 52 Hyperbola Projects ............................................................................................ 55 Chapter 4: Optimization Chapter Overview ............................................................................................. 59 The Burning Tent Problem .............................................................................. 60 The Swimming Pool Problem ......................................................................... 63 An Optimization Project .................................................................................. 66 Activity Notes Chapter 1: Ellipses ............................................................................................ 69 Chapter 2: Parabolas ......................................................................................... 78 Chapter 3: Hyperbolas ..................................................................................... 82 Chapter 4: Optimization .................................................................................. 84 Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Contents • iii Downloading Sketchpad Documents All Sketchpad documents (sketches) for Exploring Conic Sections with The Geometer’s Sketchpad are available online for download. • Go to www.keypress.com/gspmodules • Log in using your Key Online account, or create a new account and log in. • Enter this access code: ECSHS633 • A Download Files button will appear. Click to download a compressed (.zip) folder of all sketches for this book. The downloadable folder contains all of the sketches you need for this book, organized by chapter and activity. The sketches require The Geometer’s Sketchpad Version 5 software to open. Go to www.keypress.com/gsp/order to purchase or upgrade Sketchpad, or download a trial version from www.keypress.com/gsp/download. iv • Downloading Sketchpad Documents Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Introduction First, a confession: As a student, I didn’t place conic sections on my list of favorite highschool topics. The standard textbook treatment of the ellipse, parabola, and hyperbola seemed uninspired. There were messy algebraic equations with multiple square roots. There was lots of terminology. Drawing a conic meant plotting several points on graph paper and connecting them with a wobbly curve. I gave little thought to conics until I met David Dennis, then a fellow graduate student in mathematics education at Cornell University. David had a keen interest in the origins of curvedrawing devices. One historical figure in particular intrigued him: Frans van Schooten (1615–1660), a Dutch mathematician whose translation and commentary for Descartes’ landmark treatise, La Géométrie, led to the popularization of Cartesian geometry. A free online version of van Schooten’s book is available on the website of European Cultural Heritage Online. Descartes’ book was available in bookstores, but van Schooten’s work, Exercitationum Mathematicorum, Liber IV, Organica Conicarum Sectionum in Plano Descriptione (A Treatise on Devices for Drawing Conic Sections), remained tucked away in rare book collections. Luckily, Cornell’s Kroch Library contained a copy. David made the trip. When he returned from the library, David could barely contain his enthusiasm. “Go and see it for yourself,” he said. “It’s really something.” A visit to the library confirmed David’s findings. There in van Schooten’s book were drawing after drawing of devices that drew conics. The illustrations were exquisite, with artistic flourishes adorning many of them. Viewing van Schooten’s curvedrawing devices dispelled any notions that mathematical manipulatives were a modern invention. Van Schooten had beaten us all to the punch more than 350 years ago. An illustration from the title page of van Schooten’s book As I paged through van Schooten’s book, I wondered whether his ideas could find their way into today’s mathematics classrooms. His models, Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Introduction • v consisting of hinges and slotted rulers, were not always simple to build. How could students replicate them? Sketchpad provided a large part of the answer. While nothing can substitute for the experience of operating a physical model, a Sketchpad simulation comes close. In fact, Sketchpad betters van Schooten in some ways, allowing students to manipulate the parameters of any model and view their effects on a curve immediately. Not every activity in this book originates from a van Schooten device, but all share the belief that the method for producing a curve is as important as the curve itself. Customizing the Activities The activities in this collection are designed with flexible learning options in mind. Each chapter opens with an overview of the various pathways you might follow through the material. If constructing physical devices isn’t your interest, then skip to the corresponding Sketchpad models. These can be built from scratch or opened premade as described on page iv. If you’d like to understand why the various curvedrawing devices work, you can follow along with hints or strike out on your own to create an original proof. You can complete the entire section of ellipse activities or choose a set of interrelated activities spanning all three conics. Any of these choices could be a sensible option for your needs. Conic sections form the centerpiece of this book but are not the only mathematics under study. Issues of congruency, similarity, the Pythagorean theorem, trigonometry, and geometric optimization all arise naturally while studying the ellipse, parabola, and hyperbola. This book represents a marriage of 17thcentury manipulatives with 21stcentury technology. Enjoy the partnership. Acknowledgments As the preceding introduction makes clear, this book would not exist without David Dennis. His enthusiasm and knowledge of the field contributed immeasurably to this book’s content. Al Cuoco and E. Paul Goldenberg from Education Development Center, Inc. had the clever idea to use ellipses to solve optimization problems geometrically. The optimization material in this book draws from their chapter, “Dynamic Geometry as a Bridge from Euclidean Geometry to Analysis,” in Geometry Turned On (Mathematical Association of America, 1997). Nick Jackiw suggested several of the sketches in this book. He also built the nifty paraboladrawing device shown on the back cover. vi • Introduction Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Bill Finzer served as my editor for the first edition of this book. His expert guidance laid the groundwork for this revised and expanded edition. As my new editor, Steven Chanan found ways to push the conics material in new directions. It’s a much better book for his efforts. Despina Stylianou and Beth Porter provided valuable feedback on the initial draft of these materials. Finally, I thank my New York University graduate students for their help in testing these activities. Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Introduction • vii Common Commands and Shortcuts Below are some common Sketchpad actions used throughout this book. In time, these operations will become familiar, but at first you may want to keep this list by your side. To create a new sketch Choose New Sketch from the File menu. To close a sketch Choose Close from the File menu. Or click in the close box in the upperleft (Mac) or upperright (Windows) corner of the sketch. To undo or redo a recent action Choose Undo from the Edit menu. You can undo as many steps as you want, all the way back to the state your sketch was in when last opened. To redo, choose Redo from the Edit menu. To deselect everything Click in any blank area of your sketch with the Arrow tool or press Esc until objects deselect. Do this before making selections required for a command so that no extra objects are included. To deselect a single object while keeping all other objects selected, click on it with the Arrow tool. To show or hide a label Position the finger of the Text tool over the object and click. The hand will turn black when it’s correctly positioned to show or hide a label. To change a label Position the finger of the Text tool over the label and doubleclick. The letter “A” will appear in the hand when it’s correctly positioned. To change an object’s line width or color Select the object and choose from the appropriate submenu in the Display menu. To hide an object Select the object and choose Hide from the Display menu. To construct a segment’s midpoint Select the segment and choose Midpoint from the Construct menu. To construct a parallel line Select a straight object for the new line to be parallel to and a point for it to pass through. Then choose Parallel Line from the Construct menu. To construct a perpendicular line Select a straight object for the new line to be perpendicular to and a point for it to pass through. Then choose Perpendicular Line from the Construct menu. To reflect a point (or other object) Doubleclick the mirror (any straight object) or select it and choose Mark Mirror. Then select the point (or other object) and choose Reflect from the Transform menu. To trace an object Select the object and choose Trace from the Display menu. Do the same thing to toggle tracing off. (If you’d rather that traces fade, check Fade Traces Over Time on the Preferences Color panel.) To use the calculator Choose Calculate from the Number menu. To enter a measurement into a calculation, click on the measurement itself in the sketch. Keyboard shortcuts Command Mac Undo Windows Command Mac Windows Ô+Z Ctrl+Z Animate/Pause Ô+` Alt+` Redo Ô+R Ctrl+R Increase Speed Ô+] Alt+] Select All Ô+A Ctrl+A Decrease Speed Ô+[ Alt+[ Properties Ô+? Midpoint Ô+M Ctrl+M Hide Objects Ô+H Ctrl+H Intersection Ô+I Ctrl+I Segment Ô+L Ctrl+L Ctrl+P Alt+? Show/ Hide Labels Ô+K Ctrl+K Trace Objects Ô+T Ctrl+T Polygon Interior Ô+P Erase Traces Ô+B Ctrl+B Calculate viii • Common Commands and Shortcuts Ô+= Alt+= Action Mac Windows scroll drag Option+ Alt+drag drag display Context menu Control+ click rightclick navigate Toolbox Shift+arrow keys choose Arrow, deselect objects, stop animations, erase traces Esc (escape key) move selected objects 1 pixel (hold down to move continuously) , ⇥, ⇤, ⌅ keys Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press 1 Ellipses CHAPTER OVERVIEW Of all the conic sections, the ellipse is the one we see most often. Any circle, when viewed at an angle, appears elliptical. Tilt a drinking glass and the water along the surface outlines an ellipse. Place a ball on the floor and shine an angled beam of light onto it from above—the ball will cast an elliptical shadow on the floor. Slice a cone at an angle from side to side (as shown at right) and the cross section is an ellipse. (This explains why the ellipse qualifies as a conic section.) On a grander scale, the orbits of planets trace ellipses. Comets in permanent orbit around the sun follow elliptical paths. When a plane cuts only one of two cones forming a doublecone, the result is a circle or an ellipse. If the cut is perpendicular to the doublecone’s axis, the result is a circle; otherwise, it’s an ellipse. The material in this chapter falls into three general categories: • Activities I–IV introduce basic ellipse terminology (focal points, major/minor axes, eccentricity) and common ellipse construction techniques (on paper and with Sketchpad). • Activities V and VI offer two lesserknown constructions that call upon the ellipse’s distance definition for their proofs. • The constructions in activities VII and VIII highlight algebraic proofs. In total, this chapter offers thirteen ellipse constructions for you to sample. Open the multipage sketch Ellipse Tour.gsp for a handy slideshow overview. I. Getting Started: When Is a Circle Not a Circle? Introduce yourself to ellipses as a Sketchpad circle splits its center into two. Start your ellipse explorations here. II. The PinsandString Construction Swing a string (literally!) to draw an ellipse. Then use a Sketchpad model to formulate an ellipse’s distance definition. Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Chapter 1: Ellipses • 3 III. The Concentric Circles Construction Two sets of concentric circles set the stage for this classic Sketchpad ellipse construction. IV. Some Ellipse Relationships Investigate a prebuilt Sketchpad model to uncover the algebraic and geometric relationships between an ellipse’s major axis, minor axis, focal length, and eccentricity. V. The Folded Circle Construction Watch an ellipse appear before your eyes as you fold and unfold a paper circle. Model the technique with Sketchpad to reveal the underlying mathematics. VI. The Congruent Triangles Construction A pair of congruent triangles holds the key to this unusual Sketchpad ellipse construction. VII. The Carpenterʼs Construction Roll up your sleeves as you build a carpenter’s ellipsograph with a ruler and then with Sketchpad. VIII. Dannyʼs Ellipse High school student Danny Vizcaino knew there had to be a better way to construct a Sketchpad ellipse. He was right. Read all about it! IX. Ellipse Projects Round out the chapter with some enticing ellipse excursions. 4 • Chapter 1: Ellipses Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Getting Started: When Is a Circle Not a Circle? Below is a circle with center at point C. C Imagine that the circle’s center splits into two points, F1 and F2. As the points move away from each other, the circle deforms to accommodate these two “centers.” The circle is now an ellipse. F1 F2 F1 F2 F1 F2 F1 F2 F1 and F2 have special names. They’re called the focal points or foci (singular: focus) of the ellipse. Open the sketch Stretch.gsp in the Ellipse folder. You’ll see a circle . . . or is it? Drag the circle’s center and watch what happens. We draw circles by using a compass. But how can we draw ellipses? The pinsandstring activity that follows offers one possibility. Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Chapter 1: Ellipses • 5 The PinsandString Construction Name(s): There are many ways to construct an ellipse, but perhaps the most wellknown method involves just two thumbtacks and a length of string. You’ll use these materials now to build ellipses by hand. In the next activity, you’ll model a similar technique with Sketchpad. Constructing a Physical Model Preparation: You’ll need a piece of string, two thumbtacks, a corkboard, a large sheet of paper, and a pencil. 1. Place your paper on top of the corkboard and stick the two thumbtacks into it. Then tie each end of the string onto a thumbtack. Pick any locations for the thumbtacks, but make sure to leave some slack in the string. The thumbtacks will be the focal points of your ellipse. From Sive de Organica Conicarum Sectionum in Plano Descriptione, Tractatus, by Dutch mathematician Frans van Schooten, 1646. 2. Pull the string taut with your pencil. Make sure the string lies near the tip of the pencil. 3. Keeping the string taut, swing the pencil around the focal points, letting the tip of the pencil trace its path. You may need to reposition the pencil to draw the entire curve. Questions Q1 What symmetries do you see in your ellipse? Draw a picture to illustrate any lines of symmetry. Q2 Move the ends of your string so that they’re farther apart. Redraw the ellipse and describe how its shape compares to your original curve. Q3 Move the ends of your string close together. Redraw the ellipse and describe how its shape compares to your original curve. Q4 Suppose you move the ends of your string so far apart that the string is fully extended and taut. What “curve” will your pencil draw? Q5 Suppose you attach both ends of the string to the same thumbtack. What curve will your pencil draw? Q6 Suppose a friend wants to draw an ellipse identical to one of yours. What two pieces of information would you need to give her so that she could reproduce it? 6 • Chapter 1: Ellipses Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press The PinsandString Construction (continued) Q7 There are two important segments associated with every ellipse: the major axis and the minor axis. In each of the following illustrations, segment AB is the major axis, segment CD is the minor axis, and F1 and F2 are the focal points. D B F2 A F1 F2 B C D D C B F2 A F1 F1 C A Based on these pictures, write your own definition of the major and minor axes. Q8 Without using any tools, how could you form the major and minor axes of an ellipse cut out of paper? Uncover the Imposter Open the sketch Points.gsp in the Ellipse folder. You’ll see two foci, F1 and F2, of an invisible ellipse and three points, A, B, and C. The ellipse passes through exactly two of these three points, but which ones? Use Sketchpad’s measurement and calculation tools to spot the imposter. Q9 Q10 Which point doesn’t sit on the ellipse? Explain how you can tell. Complete this sentence without mentioning of pins or string: An ellipse is the set of points P such that the following value is constant for all locations of P: Explore More 1. Open the sketch Parametric Ellipses.gsp (Ellipse folder) to view ellipses constructed with Sketchpad’s parametric coloring feature. 2. Here is the definition of a new curve similar to an ellipse: The set of points P such that PA + PB + PC is constant for three fixed points, A, B, and C. Open the sketch New Curve.gsp (Ellipse folder) to see how such a curve can be constructed using Sketchpad’s parametric coloring feature. Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Chapter 1: Ellipses • 7 The Concentric Circles Construction Name(s): Circles that are concentric share the same center. In this activity, you’ll use two sets of concentric circles to draw an ellipse by hand. You’ll then transfer this technique to Sketchpad to draw an ellipse whose shape and size can be adjusted just by dragging your mouse. Sketching Ellipses by Hand The illustration below shows two sets of concentric circles. One set of circles is centered at point F1, the other at point F2. For each set, the radii of the circles increase by 1’s, from 1 unit all the way up to 6 units. Points F1 and F2 are the foci of an infinite number of ellipses, but only two that we’re interested in: the one that passes through point A and the one that passes through point B. Remember: An ellipse is the set of points P such that PF1 + PF2 is constant. Q1 How many units apart are points A and F1? How many units apart are points A and F2? What is the numerical value of AF1 + AF2? Q2 Locate and mark at least seven points that sit on the ellipse passing through point A. Explain how you found them. Q3 Locate and mark at least seven other points that sit on the ellipse passing through point B. (Use a different colored pen or pencil if possible.) Explain how you found them. Q4 Using the points you found as guidelines, sketch the two ellipses. B A F1 8 • Chapter 1: Ellipses F2 Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press The Concentric Circles Construction (continued) Constructing a Sketchpad Model Now that you’ve drawn some ellipses by hand, it’s time to build a dynamic one: an ellipse that changes shape as its parts are dragged. Follow the steps below to construct a Sketchpad ellipse. As you do, think about how this method relates to the concentric circles technique. A F1 C B F2 1. Draw a horizontal line near the top of your screen. Hide the control points by selecting them and choosing Hide Points from the Display menu. 2. Construct points A, B, and C on the line. The location of these points doesn’t matter, but make sure point C is between points A and B. 3. Hide the line. Draw individual segments AC and CB. 4. Draw points F1 and F2 to represent the foci of your ellipse. 5. Using the Circle by Center+Radius command, construct a circle with center F1 and radius AC. Construct another circle with center F2 and radius CB. 6. Construct the intersection points of the circles. (You might have to adjust your model so the circles intersect.) 7. Select the intersection points and choose Trace Intersections from the Display menu. If you don’t want your traces to fade, be sure the Fade Traces Over Time box is unchecked on the Color panel of the Preferences dialog box. 8. Drag point C back and forth slowly along AB and observe the trace of the two points. 9. Change the distance between F1 and F2. Then, if necessary, choose Erase Traces from the Display menu to erase your previous curve. Trace several new curves, each time varying the distance between the focal points. Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Chapter 1: Ellipses • 9 The Concentric Circles Construction (continued) For every new location of F1 and F2, you need to retrace your curve. Ideally, your ellipse should adjust automatically as you drag either focus. Sketchpad’s powerful Locus command makes this possible. 10. Turn tracing off for the two intersection points by selecting them and once again choosing Trace Intersections from the Display menu. 11. Select one of the two intersection points and point C. Choose Locus from the Construct menu. Do this again for the other intersection point and point C. You’ll form an entire curve: the locus of the two intersection points as point C moves along segment AB. Drag F1, F2, or point B to change the size and shape of the ellipse. Questions Q5 As you drag point C along segment AB, the radii of both circles change lengths. Still, there is a relationship that exists between the two radii regardless of point C’s position. What is it? Q6 Explain why the two intersection points of the two circles trace an ellipse. Q7 How far apart can the two focal points be before you can no longer trace an ellipse? Q8 Select your two onscreen circles, choose Trace Circles from the Display menu, then drag point C along segment AB. Based on this experiment, describe the similarities between your Sketchpad construction and the concentric circles technique. Explore More 1. By shortening the distance between points A and B and dragging point C to the left of A and to the right of B so that it does not lie between them, it’s possible to draw a different type of curve. Try it and see what you get. 2. Consider the definition of a constantperimeter rectangle: A constantperimeter rectangle is a rectangle constructed in Sketchpad whose dimensions can change, but whose perimeter always remains fixed at a given, constant value. This means that a rectangle with constant perimeter of 20 inches could have dimensions of 3”⇥ 7”, 6”⇥ 4”, or 2”⇥ 8”, but not 5”⇥ 6”. Use the techniques you learned when building a Sketchpad ellipse to construct a constantperimeter rectangle. 10 • Chapter 1: Ellipses Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Some Ellipse Relationships Name(s): In the PinsandString activity, you drew ellipses with a pencil and a taut piece of string. Now, you’ll use Sketchpad to explore some of the algebraic and geometric relationships that exist between this string and an ellipse’s major and minor axes. Finding Lengths Open the sketch String.gsp in the Ellipse folder. You’ll see an ellipse with major axis AB, minor axis CD, and focal points at F1 and F2. Point P represents the pencil point that’s pulling taut a string attached to points F1 and F2. Together, segments PF1 and PF2 represent the string. The total length of the “string” is 20 units (PF1 + PF2). The distance between F1 and F2 is 16 units. D P A F1 F2 B C Questions Answer these questions without taking any measurements with Sketchpad. For each one, drag point P around the ellipse until you find a location that helps you answer the question. Q1 What is the length of the major axis AB? Explain where you positioned point P to reach your conclusion. Q2 What is the length of the minor axis CD? Explain where you positioned point P to reach your conclusion. Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Chapter 1: Ellipses • 11 Some Ellipse Relationships (continued) Eccentricity Some ellipses are “skinny” and elongated. Others are “fat” and nearly circular. The eccentricity of an ellipse is a measure that captures the shape of an ellipse in one numerical value. The eccentricity of an ellipse is defined as: the distance between the focal points the distance between the endpoints of the major axis 1. Open the sketch Eccentricity.gsp in the Ellipse folder. You’ll see an ellipse with major axis endpoints A and B, and focal points F1 and F2. 2. Use Sketchpad’s Distance command to measure the two distances needed to compute the ellipse’s eccentricity. 3. Use Sketchpad’s Calculate command to compute the eccentricity. Questions Q3 Your Sketchpad ellipse can change its size and shape to represent a whole collection of ellipses. Before experimenting with it, make a prediction: How small and how large do you think an ellipse’s eccentricity can become? Explain your reasoning. Q4 Drag point B to change the size and shape of the ellipse. As you do so, monitor the values of its eccentricity. Does your prediction from Q3 hold? Modify it if necessary. Q5 How many ellipses can share the same eccentricity? How could you create two ellipses with the same eccentricity without using Sketchpad? Explore More See Q2 from earlier in the activity for help with this construction. 1. Open the second page of the sketch String.gsp. You’ll see a new ellipse along with the length of “string” used to draw it. Construct the focal points of the ellipse. No measuring allowed! Make sure your foci are dynamic: they should adjust themselves to remain in the proper locations as you change the length of the string. 2. Johannes Kepler’s First Law of planetary motion states that the orbit of each planet is an ellipse with the Sun at one focus. Do some research to find the eccentricities of our planets. Why might astronomers before Kepler have believed planets moved in circular motion? 12 • Chapter 1: Ellipses Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press The Folded Circle Construction Name(s): Sometimes a conic section appears in the unlikeliest of places. In this activity, you’ll explore a paperfolding construction in which crease lines interact in a surprising way to form a conic. Constructing a Physical Model Preparation: Use a compass to draw a circle with a radius of approximately three inches on a piece of wax paper or patty paper. Cut out the circle with a pair of scissors. (If you don’t have these materials, you can draw the circle in Sketchpad and print it.) 1. Mark point A, the center of your circle. If you’re working in a class, have members place B at different distances from the center. If you’re working alone, do this section twice— once with B close to the center, once with B close to the edge. 2. Mark a random point B within the interior of your circle. 3. As shown below right, fold the circle so that a point on its circumference lands directly onto point B. Make a sharp crease to keep a record of this fold. Unfold the circle. B A 4. Fold the circle along a new crease so that a different point on the circumference lands on point B. Unfold the circle and repeat the process. 5. After you’ve made a dozen or so creases, examine them to see if you spot any emerging patterns. Mathematicians would describe your set of creases as an envelope of creases. B A 6. Resume creasing your circle. Gradually, a welloutlined curve will appear. Be patient— it may take a little while. 7. Discuss what you see with your classmates and compare their folded curves to yours. If you’re doing this activity alone, fold a second circle with point B in a different location. Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Chapter 1: Ellipses • 13 The Folded Circle Construction (continued) Questions Q1 The creases on your circle seem to form the outline of an ellipse. What appear to be its focal points? Q2 If you were to move point B closer to the edge of the circle and fold another curve, how do you think its shape would compare to the first curve? Q3 If you were to move point B closer to the center of the circle and fold another curve, how do you think its shape would compare to the first curve? Constructing a Sketchpad Model Fold and unfold. Fold and unfold. Creasing your circle takes some work. Folding one or two sheets is fun, but what would happen if you wanted to continue testing different locations for point B? You’d need to keep starting with fresh circles, folding new sets of creases. Sketchpad can streamline your work. With just one circle and one set of creases, you can drag point B to new locations and watch the crease lines adjust themselves instantaneously. C 8. Open a new sketch and use the Compass tool to draw a large circle with center A. Hide the circle’s radius point. B 9. Use the Point tool to draw a point B at a random spot inside the circle. A crease 10. Construct a point C on the circle’s circumference. 11. Construct the “crease” formed when point C is folded onto point B. 12. Drag point C around the circle. If you constructed your crease line correctly, it should adjust to the new locations of point C. 13. Select the crease line and choose Trace Line from the Display menu. If you don’t want your traces to fade, be sure the Fade Traces Over Time box is unchecked on the Color panel of the Preferences dialog box. 14. Drag point C around the circle to create a collection of crease lines. 15. Drag point B to a different location and then, if necessary, choose Erase Traces from the Display menu. 16. Drag point C around the circle to create another collection of crease lines. 14 • Chapter 1: Ellipses Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press The Folded Circle Construction (continued) Retracing creases for each location of point B is certainly faster than folding new circles. But we can do better. Ideally, your crease lines should relocate automatically as you drag point B. Sketchpad’s powerful Locus command makes this possible. 17. Turn tracing off for your original crease line by selecting it and once again choosing Trace Line from the Display menu. 18. Now select your crease line and point C. Choose Locus from the Construct menu. An entire set of creases will appear: the locus of crease locations as point C moves along its path. If you drag point B, you’ll see that the crease lines readjust automatically. 19. Save your sketch as Creased Circle.gsp. You’ll use it again in one of the Hyperbola Projects activities. Questions The Merge and Split commands appear in the Edit menu. Q4 How does the shape of the curve change as you move point B closer to the edge of the circle? Q5 How does the shape of the curve change as you move point B closer to the center of the circle? Q6 Select point B and the circle. Then merge point B onto the circle’s circumference. Describe the crease pattern. Q7 Select point B and split it from the circle’s circumference. Then merge it with the circle’s center. Describe the crease pattern. Playing Detective Each crease line on your circle touches the ellipse at exactly one point. Another way of saying this is that each crease is tangent to the ellipse. By engaging in some detective work, you can locate these tangency points and use them to construct just the ellipse without its creases. 20. Open the sketch Folded Circle.gsp in the Ellipse folder. You’ll see a thick crease line and its locus already in place. 21. Drag point C and notice that the crease line remains tangent to the ellipse. The exact point of tangency lies at the intersection of two lines—the crease line and another line not shown here. Construct this line in your sketch as well as the point of tangency, point E. Select the locus and make its width thicker so that it’s easier to see. 22. Select point E and point C and choose Locus from the Construct menu. If you’ve identified the tangency point correctly, you should see a curve appear precisely in the white space bordered by the creases. Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Chapter 1: Ellipses • 15 The Folded Circle Construction (continued) How to Prove It C The Folded Circle construction seems to generate ellipses. Can you prove that it does? Try developing a proof on your own, or work through the following steps and questions. The picture at right should resemble your construction. Line HI (the perpendicular bisector of segment CB) represents the crease formed when point C is folded onto point B. Point E sits on the curve itself. I E B A H 23. Add segments CB, BE, and AC to the picture. 24. Label the intersection of CB with the crease line as point D. Questions Remember: An ellipse is the set of points such that the sum of the distances from each point to two fixed points (the foci) is constant. Q8 Use a triangle congruence theorem to prove that jBED m jCED. Q9 Segment BE is equal in length to which other segment? Why? Q10 Use the distance definition of an ellipse and the result from Q9 to prove that point E traces an ellipse. Explore More 1. When point B lies within its circle, the creases outline an ellipse. What happens when point B lies outside its circle? 2. Use the illustration from your ellipse proof to show that ⇧AEH = ⇧BED. Here’s an interesting consequence of this result: Imagine a pool table in the shape of an ellipse with a hole at one of its focal points. If you place a ball on the other focal point and hit it in any direction without spin, the ball will bounce off the side and go straight into the hole. Guaranteed! 16 • Chapter 1: Ellipses Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press The Congruent Triangles Construction A linkage is any device with hinged and slotted rods. Name(s): The picture below appears in a book by the seventeenthcentury mathematician Frans van Schooten. It shows a linkage consisting of three movable rods hinged together. In this activity, you’ll explore a Sketchpad model of van Schooten’s device and prove that it draws ellipses. 1. Open the sketch Congruent Triangles.gsp in the Ellipse folder. You’ll see a construction that matches most of van Schooten’s picture. This sketch was created so that two equalities always hold. These are: AB = FC BF = CA C F G E A B 2. Drag point F. As you do, observe how the colorcoded segments remain fixed in size and equal to each other. 3. Select point E and choose Trace Intersection from the Display menu. Drag point F and observe the curve traced by point E. 4. You’ve drawn what looks like half an ellipse. To trace the other half, use Sketchpad’s Reflect command to reflect point E across segment AB. Select the reflected point, E , and choose Trace Point. Now drag point F to trace the entire curve. Questions Q1 What appear to be the focal points of your ellipse? Q2 The blue segment at the bottom left of your screen controls the lengths of segments AB and FC. Drag its right endpoint to lengthen or shorten it. Choose Erase Traces from the Display menu to remove your previous curve. Now, retrace your curve. Do this several times. How does the length of the blue segment affect your curve? Q3 If you look at van Schooten’s picture, you’ll see that it includes a point G. Construct lines through segments AB and FC on your sketch so they meet at point G. Now draw a line through points E and G. Drag point F. What is the relationship between this newly created line and your curve? Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Chapter 1: Ellipses • 17 The Congruent Triangles Construction (continued) How to Prove It It certainly looks like the Congruent Triangles construction draws ellipses, but can you explain why? Try developing a proof on your own, or work through the following questions. Questions Q4 Add segment AF to your sketch. Using a triangle congruence theorem, show that jABF m jFCA. Q5 Use the result from the question above to complete this statement: ⇧FCA = ⇧ Remember: An ellipse is the set of points such that the sum of the distances from each point to two fixed points (the foci) is constant. Q6 Use the angle equality above and other information you know about the linkage to show that jAEB m jFEC. Q7 Segment AE is equal in length to which other segment? Why? Q8 Use the distance definition of an ellipse and the result from Q7 to prove that point E traces an ellipse. Explore More 1. This construction is sometimes called the crossed parallelogram. Explain why. 2. Use Sketchpad’s Trace command to display the locus of point C as you drag it. Describe all of the similarities you can find between this sketch and the Folded Circle construction. You’ll need to make liberal use of Sketchpad’s Circle by Center+Radius command. 3. Open the sketch Gears.gsp in the Ellipse folder to operate a pair of elliptic gears. 4. Starting with a blank sketch, build your own Sketchpad model of the Congruent Triangles construction. 18 • Chapter 1: Ellipses Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press The Carpenter’s Construction Name(s): The device pictured at right, a favorite among carpenters and woodworkers, is called an ellipsograph or trammel. The ellipsograph first appears in the work of Proclus (A.D. 410–485). An ellipsograph has an arm with two bolts that slide along a pair of perpendicular tracks. As the bolts glide along their respective grooves, a pen attached to one end draws an ellipse. In this activity, you’ll build your own ellipsograph with a ruler, then explore a more robust model with Sketchpad. Finally, you’ll prove that this device really does draw ellipses. Constructing a Physical Model Preparation: You’ll need a ruler, some masking tape, a pen or pencil, and a large piece of paper (11”⇥ 17” is good, but 8.5” ⇥ 11” works also). If you’re working in a class, have members pick different placements of B relative to A and C. If you’re working alone, do this activity twice with different point B locations. 1. Put a long strip of masking tape on a ruler, lining up an edge of the tape with an edge of the ruler. Mark three points, A, B, and C, on the tape. Make sure the distance between points A and C is less than half the width of your paper. A B C 2. Use your ruler to draw a pair of perpendicular lines on the paper. The illustration on the next page shows how to slide your ruler along the lines. 3. Begin by positioning point B at the intersection of the two lines and point A on the horizontal line to the left of B. Place a mark on your paper at point C (also on the horizontal line for now). 4. Slide the ruler just a little so that point A continues to lie on the horizontal line and point B lies on the vertical line. Mark the location of point C. 5. Continue to slide points A and B in small increments, keeping point A on the horizontal line and point B on the vertical line. Each time you reposition the ruler, mark point C’s position. Eventually the ruler will sit vertically with point A at the intersection point. Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Chapter 1: Ellipses • 19 The Carpenterʼs Construction (continued) 6. You’ve drawn what appears to be a quarter ellipse. Continue repositioning the ruler to draw the entire curve. 7. Discuss what you see with your classmates and compare their curves to yours. If you’re doing this activity alone, draw a new curve by changing the location of point B. C A B C A B C B A Questions Q1 Shown below are two rulers with different relative locations for points A, B, and C. If each ruler is used to draw an ellipse, how will the shape of the two curves differ? A A B C B C Q2 Assuming that ellipsographs do draw ellipses, how would you position points A, B, and C to draw an ellipse with a 20cm major axis and a 12cm minor axis? Q3 Can an ellipsograph draw circles? Explain why or why not. 20 • Chapter 1: Ellipses Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press The Carpenterʼs Construction (continued) Investigating a Sketchpad Model Your ruler model of an ellipsograph provides a good sense of how the device works. Yet sliding the ruler bit by bit was probably awkward. A Sketchpad ellipsograph offers some advantages: a smooth, continuous motion and easily adjustable lengths. 8. Open the sketch Carpenter.gsp in the Ellipse folder. C 9. Drag point A. As you do, watch the motion of segment AC. B 10. Select point C and choose Trace Intersection from the Display menu. Drag point A and observe the curve traced by point C. The curve you see is the locus of point C as point A moves along its line. A 11. You’ve drawn what looks like half an ellipse. To trace the other half, use Sketchpad’s Reflect command to reflect point C across the horizontal axis. Select the reflected point, C , and choose Trace Point. Now drag point A to trace the entire curve. 12. Adjust the lengths of AB and BC at the bottom left of your screen to vary the parameters of the ellipsograph. Then choose Erase Traces from the Display menu to erase your previous curve. Trace several new curves, each time varying the parameters AB and BC. 13. Turn tracing off for points C and C by selecting them and once again choosing Trace Points from the Display menu. 14. Now select points A and C. Choose Locus from the Construct menu. Do this again for points A and C . You’ll form an entire curve: the locus of points C and C . 15. As before, adjust the lengths of AB and BC to observe their effect on the curve. Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Chapter 1: Ellipses • 21 The Carpenterʼs Construction (continued) How to Prove It The curves you’ve drawn certainly look like ellipses. But appearances don’t always tell the whole story. A proof can confirm the curves’ identities and provide insights into the mathematics underlying ellipsographs. Do you know the algebraic representation of an ellipse? Take a look at the following definition: The points satisfying the equation x2 a2 + y2 b2 = 1, with b > a, lie on an ellipse centered at the origin with major axis of length 2b along the yaxis and minor axis of length 2a along the xaxis. The illustration below shows an ellipsograph in the xy plane whose arm is represented by segment AC. For this particular ellipsograph, AB = 6 and BC = 3. Several extra segments are included in the picture: segment CE is parallel to the yaxis and segment BD is parallel to the xaxis. Since the location of point C changes as the ellipsograph’s arm moves, it’s labeled as (x, y), using variables as coordinates. If C traces an ellipse, you should be able to derive an equation like the one above relating x to y. Questions The questions that follow provide a stepbystep guided proof. You can answer them or write your own proof without any hints. C = (x, y) 3 B D 6 A 22 • Chapter 1: Ellipses (0, 0) E Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press The Carpenterʼs Construction (continued) Hint: Determine the lengths of the ellipse’s major and minor axes, and the location of its center. Q4 Assume for a moment—without proof—that the ellipsograph in the picture on the previous page draws an ellipse. Given that AB = 6 and BC = 3, what is its equation? (We’ll compare our eventual answer with this equation later.) Q5 Fill in the lengths of the following segments in terms of x and y: BD = CE = CD = Explain how you found the length of CD. Q6 Now that you’ve determined the lengths of various segments, find a way to relate x to y. Here are two approaches you might consider: • Look for a pair of similar triangles in the diagram. Use their similarity to create a proportion relating x to y. • Compute sin(⇧CAE) and cos(⇧CBD). Look for a way to relate these two values to each other. If necessary, manipulate your equation so that it’s recognizable as that of an ellipse. Compare your equation to the one you found in Q4 to see if they match. Q7 Rewrite your proof, this time making it more general. Let AB = s and BC = t. Explore More 1. Starting with a blank sketch, build your own Sketchpad model of the ellipsograph construction. 2. Given any curve drawn by an ellipsograph, you should be able to find its foci. Open page 3 of the sketch Carpenter.gsp. Use what you’ve learned in the Some Ellipse Relationships activity to construct the foci of the ellipse on screen. Make sure your foci are dynamic: they should adjust to remain in the proper locations as you change the lengths of segments AB and BC. Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Chapter 1: Ellipses • 23 The Carpenterʼs Construction (continued) For added drama, imagine that you’re standing on the ladder. What path does your foot trace? 3. The illustrations below show a seventeenthcentury drawing device whose motion resembles that of a ladder sliding down a wall. On the ladder sits a bucket. As the ladder slides, what path does the bucket trace? Investigate this problem by modifying your physical model of the ellipsograph. Build a Sketchpad model, too. Can you prove your findings? 4. There’s a theorem from geometry that states: The midpoint of the hypotenuse of a right triangle is equidistant from the three vertices of the triangle. Assume for the moment that this statement is true. How can you use it (and nothing else) to prove that the bucket from Q3 traces a circle when it’s midway up the ladder? 5. Leonardo da Vinci devised an ellipsetracing technique that substitutes a sliding triangle for the sliding ellipsograph. C B Open the sketch Triangle.gsp in the Ellipse folder. Examine the path traced by triangle vertex C as the other two vertices slide along the x and yaxes. 24 • Chapter 1: Ellipses A Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Danny’s Ellipse Name(s): While a student at Mountain View High School in California, Danny Vizcaino devised a novel way to construct ellipses with Sketchpad. In this activity, you’ll build a model of Danny’s ellipse, then prove why his method works. Constructing a Sketchpad Model 1. Open a new sketch. Choose Show Grid from the Graph menu. Then choose Hide Grid to remove the grid lines while keeping the x and yaxes. D B E F 2. Label the origin (the intersection of the two lines) as point A. 3. Draw a random point B on the yaxis and a point C on the xaxis. A C c1 c2 4. Select, in order, points A and B. Then choose Circle by Center+Point from the Construct menu to build a circle c1 with center at point A passing through point B. 5. Repeat step 4 to construct a circle c2 with center at point A passing through point C. 6. Draw a segment from point A to a random point D on circle c2. 7. Construct point E, the intersection of segment AD with circle c1. 8. Construct a line through point D perpendicular to the xaxis. 9. Construct a line through point E perpendicular to the yaxis. 10. Construct point F, the intersection of the two lines you just created. If you don’t want your traces to fade, be sure the Fade Traces Over Time box is unchecked on the Color panel of the Preferences dialog box. 11. Select point F and choose Trace Intersection from the Display menu. Drag point D around its circle and observe the curve traced by point F. The curve you see is the locus of point F as point D moves around its circle. Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Chapter 1: Ellipses • 25 Dannyʼs Ellipse (continued) 12. Drag points B and C to alter the sizes of circles c1 and c2. Then, if necessary, choose Erase Traces from the Display menu to erase your previous curve. Trace several new curves, each time varying the sizes of the two circles. For every new location of points B and C, you need to retrace your curve. Ideally, your ellipse should adjust itself automatically. Sketchpad’s powerful Locus command makes this possible. 13. Turn tracing off for point F by selecting it and once again choosing Trace Intersection from the Display menu. 14. Now select points D and F. Choose Locus from the Construct menu. You’ll form an entire curve: the locus of point F. Drag points B and C to vary the shape of the curve. How to Prove It Open Dynamic Geometry.gsp in the Ellipse folder to see a fun application of Danny’s method. Intrigued by Danny’s Sketchpad construction, Key Curriculum Press sponsored a worldwide contest to answer the following question: Danny’s curve looks like an ellipse, but is it? The contest is over, but the challenge remains. Can you prove that Danny’s curve is an ellipse? To do so, you’ll need the algebraic definition below. The points satisfying the equation x2 a2 + y2 b2 = 1, with a > b, lie on an ellipse centered at the origin with major axis of length 2a along the xaxis and minor axis of length 2b along the yaxis. In the illustration on the next page, AE = 2 and ED = 3. Segment EH is perpendicular to the xaxis. Since the location of point F changes as point D moves, it’s labeled as (x, y), using variables as coordinates. If F traces an ellipse, you should be able to derive an equation like the one above relating x to y. 26 • Chapter 1: Ellipses Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Dannyʼs Ellipse (continued) Questions The questions that follow provide a stepbystep guided proof. You can answer them or write your own proof without any hints. Q1 Assume for a moment—without proof—that Danny’s curve is an ellipse. What is its equation? (We’ll compare our eventual answer with this equation later.) Hint: Determine the lengths of the ellipse’s major and minor axes and the location of its center. For Q1 and Q2, use the numerical values in the picture: AE = 2 and ED = 3. Q2 D 3 E F = (x, y) 2 Fill in the lengths of the following segments in terms of x and y: A H G c1 AG = EH = c2 DG = Explain how you found the length of DG . Q3 Now that you’ve determined the lengths of various segments, find a way to relate x to y. Here are two approaches you might consider: • Look for a pair of similar triangles in the diagram. Use their similarity to create a proportion relating x to y. • Compute sin(⇧EAH) and cos(⇧DAG). Look for a way to relate these two values to each other. If necessary, manipulate your equation so that it’s recognizable as that of an ellipse. Compare your equation to the one you found in Q1 to see if they match. Q4 Rewrite your proof, this time making it more general. Let AE = s and ED = t. Explore More See page 3 of Danny.gsp for help. 1. Since Danny’s curve is an ellipse, you should be able to find its foci. Open the second page of the sketch Danny.gsp in the Ellipse folder. Use what you’ve learned in the Some Ellipse Relationships activity to construct the foci of the ellipse on screen. Make sure your foci are dynamic: they should adjust to remain in the proper locations as you drag point B or C. Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Chapter 1: Ellipses • 27 Ellipse Projects The projects below extend your ellipse knowledge in new directions and are ideal for inclass presentations. 1. Recall the distance definition of an ellipse: An ellipse is the set of point P such that PA + PB is constant for two fixed points, A and B. Suppose we define a new curve with a similar description: The set of points P such that PA + 2PB is constant for two fixed points, A and B. What does this curve look like? Build a Sketchpad model by modifying the Concentric Circles construction. 2. In the Folded Circle construction, you built the crease line formed when point C is folded onto point B by constructing the perpendicular bisector of segment BC. Imagine now that Sketchpad’s Midpoint and Perpendicular Line commands are broken. How can you construct the crease line without them? The picture below offers one possibility. It’s from the seventeenthcentury mathematician Frans van Schooten and shows a rhombus FCGB with a slotted rod passing through points F and G. Open the sketch Rhombus.gsp in the Ellipse folder and experiment with the model. How is this model similar to the Folded Circle construction? What purpose does the rhombus serve? C G E F A 28 • Chapter 1: Ellipses B Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Ellipse Projects (continued) 3. The book Feynman’s Lost Lecture: The Motion of Planets Around the Sun (W. W. Norton & Company, 2000) features a lecture by legendary physicist Richard Feynman. In his talk, Feynman uses the Folded Circle construction to demonstrate geometrically that planets orbit the sun in elliptic paths. Read Feynman’s lecture and prepare a report on his method. 4. Build your own working model of the Congruent Triangles construction using cardboard and paper fasteners. What happens when AB ⌅ BC ? 5. Open the sketch Bent Straw.gsp in the Ellipse folder. You’ll see a linkage with equallength segments AB and BC. As you drag point C, notice how the motion of this device resembles that of a bent straw. Can you prove point D traces an ellipse? The sketch contains some suggestions to get you started. B D A C You’ll use this tool in the Burning Tent activity later in this book. 6. Use techniques from the Some Ellipse Relationships activity as well as from Danny’s Ellipse to build a custom tool that takes three points—A, B, and P—and constructs an ellipse passing through P with A and B as foci. If you need help, the sketch Foci and Point.gsp in the Ellipse folder provides assistance. 7. The sketch Tangent Circles.gsp in the Ellipse folder shows a red circle c3 that’s simultaneously tangent to circles c1 and c2. Press the Animate button and observe the path of point C, the center of circle c3. Can you prove that C traces an ellipse? Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Chapter 1: Ellipses • 29 2 Parabolas CHAPTER OVERVIEW The path of a baseball. The curve formed by the cables on the Golden Gate Bridge. The trail of water jetting out from a hose. All of these are examples of parabolic curves (or very nearly so). The picture at right shows a cone that’s been sliced by a plane parallel to a side. The cross section (the “conic section”) is a parabola. This chapter opens with a look at a parabola’s focus and directrix (activity I). It then presents two parabola constructions, one based on the parabola’s distance definition (activity II) and the other on a parabola’s algebraic form (activity III). When a doublecone is cut by a plane that’s parallel to an edge, the result is a parabola. In total, this chapter offers seven parabola constructions for you to sample. Open the multipage sketch Parabola Tour.gsp for a handy slideshow overview. I. Introducing the Parabola A focal point and a directrix line are the main ingredients for two parabola constructions. II. The Folded Rectangle Construction With just a blank sheet of paper and a single point, you can fold yourself a genuine parabola. Model the technique with Sketchpad to reveal the underlying mathematics. III. The Expanding Circle Construction As a circle grows and shrinks, it defines points that lie along a parabola. Investigate this tenthcentury construction with the aid of Sketchpad. IV. Parabola Projects Round out the chapter with some pleasing parabola projects. Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Chapter 2: Parabolas • 33 Introducing the Parabola Name(s): In this activity, the geometric definition of a parabola serves as a gateway for investigating two different ways to construct the curve. Defining a Parabola Below is the geometric definition of a parabola: A parabola is the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix). If you open the sketch Parabola.gsp in the Parabola folder, you’ll see a parabola along with its focus and directrix. The sketch also contains two measurements showing the distance of point P from the focus and from the directrix. Drag point P. You’ll see the distance measurements change, but always remain equal to each other. The parabola is a symmetric curve. Its line of symmetry passes through the focus and through a point called the vertex. In the parabola below, the vertex is the lowest point on the curve. focus vertex directrix Questions Alternatively, press the show segments button. Q1 Given a line d and a point P not on the line, how do we define the distance between them? Q2 Draw a segment from point P to the focus. Then construct a segment whose length represents the distance from point P to the directrix. The segments should adjust themselves as point P moves along the parabola. Use Sketchpad to measure the lengths of these two segments. What do you expect to find? 34 • Chapter 2: Parabolas Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Introducing the Parabola (continued) Q3 Given just a parabola’s focus and directrix, how can you construct its vertex? The Concentric Circles Method Concentric circles share the same center. The illustration below shows nine concentric circles centered at point A. The radii of the circles increase by 1’s, from 1 unit all the way up to 9 units. The horizontal lines are also spaced 1 unit apart. Each line, except the one passing through point A, is tangent to a circle. We can use this arrangement to draw parabolas. B 4 3 A 2 1 Questions Q4 How many units apart are points A and B? How many units apart are point B and line 1? Based on these measurements, what can you conclude? Q5 Locate and mark at least 15 points (including the vertex) that sit on a parabola with focal point at A and line 1 as its directrix. Explain how you found them. Q6 Using the points you found as guidelines, sketch the parabola. Q7 Repeat the previous two questions, drawing parabolas with directrix lines 2, 3, and 4. All three parabolas will have A as their focal point. For each parabola, use a different colored pen or pencil if possible. Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Chapter 2: Parabolas • 35 Introducing the Parabola (continued) The Sliding Ruler Method Open the sketch Sliding Ruler.gsp in the Parabola folder. You’ll see the model below. The rectangles represent two rulers. A piece of string equal in length to BD is attached from point A to the corner of the vertical ruler (point B). The string is held taut against the edge of the ruler by a pencil at point C. Sliding ruler BD to the right while keeping the string taut causes point C to trace half a parabola. To operate the Sketchpad model, drag point D. B C A D How to Prove It The Sliding Ruler construction seems to draw parabolas. Can you prove that it does? Try developing a proof on your own or work through the following questions. Questions Remember: The length of string is equal to the length of ruler BD. Q8 Assuming the pencil at point C traces a parabola, where are the focus and directrix? Q9 Assuming the pencil at point C traces a parabola, which two segments must you prove equal in length? Q10 Complete this statement: BC + CA = BC + Q11 36 • Chapter 2: Parabolas Complete the proof. Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Introducing the Parabola (continued) Explore More 1. Circles come in different sizes, but all circles share the same shape. Given any two circles, you can enlarge or reduce one circle on a photocopy machine so that it matches the other. Believe it or not, all parabolas possess the same property: Given any two, it’s possible to enlarge or reduce one parabola on a photocopy machine so that it matches the other. Not convinced? Open the sketch Scale.gsp in the Parabola folder. Two parabolas appear onscreen—one red, one blue. Drag the unit point “1” on the xaxis. The blue parabola will stay in place, but the red parabola will adjust to the change of scale on the x and yaxes. Because the scaling on the two axes grows and shrinks in unison, the equation of the red parabola (y = x2) doesn’t change. By dragging the unit point, you should be able to make the two parabolas overlap. Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Chapter 2: Parabolas • 37 The Folded Rectangle Construction Name(s): With nothing more than a sheet of paper and a single point on the page, you can create a parabola. No rulers and no measuring required! Constructing a Physical Model Preparation: You’ll need a rectangular or square piece of wax paper or patty paper. If you don’t have these materials, use a plain sheet of paper. If you’re working in a class, have members place A at different distances from the edge. If you’re working alone, do this section twice— once with A close to the edge, once with A farther from the edge. 1. Mark a point A approximately one inch from the bottom of the paper and centered between the left and right edges. A 2. As shown below right, fold the paper so that a point on the bottom edge lands directly onto point A. Make a sharp crease to keep a record of this fold. Unfold the crease. 3. Fold the paper along a new crease so that a different point on the bottom edge lands on point A. Unfold the crease and repeat the process. A 4. After you’ve made a dozen or so creases, examine them to see if you spot any emerging patterns. Mathematicians would describe your set of creases as an envelope of creases. 5. Resume creasing the paper. Gradually, you should see a welloutlined curve appear. Be patient—it may take a little while. 6. Discuss what you see with your classmates and compare their folded curves to yours. If you’re doing this activity alone, fold a second sheet of paper with point A farther from the bottom edge. Questions Q1 The creases on your paper seem to form the outline of a parabola. Where do its focus and directrix appear to be? Q2 If you were to move point A closer to the bottom edge of the paper and fold another curve, how do you think its shape would compare to the first curve? 38 • Chapter 2: Parabolas Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press The Folded Rectangle Construction (continued) Constructing a Sketchpad Model Fold and unfold. Fold and unfold. Creasing your paper takes some work. Folding one or two sheets is fun, but what would happen if you wanted to continue testing many different locations for point A? You’d need to keep starting over with fresh paper, folding new sets of creases. Sketchpad can streamline your work. With just one set of creases, you can drag point A to new locations and watch the crease lines adjust themselves instantaneously. 7. Open a new sketch. Use the Line tool to draw a horizontal line near the bottom of the screen. This line represents the bottom edge of the paper. 8. Draw a point A above the line, roughly centered between the left and right edges of the screen. A crease B 9. Construct a point B on the horizontal line. 10. Construct the “crease” formed when point B is folded onto point A. 11. Drag point B along its line. If you constructed your crease line correctly, it should adjust to the new locations of point B. If you don’t want your traces to fade, be sure the Fade Traces Over Time box is unchecked on the Color panel of the Preferences dialog box. 12. Select the crease line and choose Trace Line from the Display menu. 13. Drag point B along the horizontal line to create a collection of crease lines. 14. Drag point A to a different location, then, if necessary, choose Erase Traces from the Display menu. 15. Drag point B to create another collection of crease lines. Retracing creases for each location of point A is certainly faster than folding paper. But we can do better. Ideally, your crease lines should relocate automatically as you drag point A. Sketchpad’s powerful Locus command makes this possible. 16. Turn tracing off for your original crease line by selecting it and once again choosing Trace Line from the Display menu. 17. Now select your crease line and point B. Choose Locus from the Construct menu. An entire set of creases will appear: the locus of crease locations as point B moves along its path. If you drag point A, you’ll see that the crease lines readjust automatically. Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Chapter 2: Parabolas • 39 The Folded Rectangle Construction (continued) Questions Q3 How does the appearance of the curve change as you move point A closer to the horizontal line? Q4 How does the appearance of the curve change as you move point A away from the horizontal line? Playing Detective Each crease line on your paper touches the parabola at exactly one point. Another way of saying this is that each crease is tangent to the parabola. By engaging in some detective work, you can locate these tangency points and use them to construct just the parabola without its creases. 18. Open the sketch Folded Rectangle.gsp in the Parabola folder. You’ll see a thick crease line and its locus already in place. 19. Drag point B and notice that the crease line remains tangent to the parabola. The exact point of tangency lies at the intersection of two lines—the crease line and another line not shown here. Construct this line in your sketch as well as the point of tangency, point D. Select the locus and make its width thicker so that it’s easier to see. 20. Select point D and point B and choose Locus from the Construct menu. If you’ve constructed point D correctly, you should see a curve appear precisely in the white space bordered by the creases. How to Prove It The Folded Rectangle construction seems to generate parabolas. Can you prove that it does? Try developing a proof on your own or work through the following steps and questions. The picture below should resemble your Sketchpad construction. Line EF (the perpendicular bisector of segment AB) represents the crease line formed when point B is folded onto point A. Point D sits on the curve itself. G F D A C E 40 • Chapter 2: Parabolas B Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press The Folded Rectangle Construction (continued) Questions Remember, a parabola is the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix). Q5 Assuming point D traces a parabola, which two segments must you prove equal in length? Q6 Use a triangle congruence theorem to prove that jACD m jBCD. Q7 Use the distance definition of a parabola and the result from Q6 to prove that point D traces a parabola. Explore More 1. Open the sketch Tangent Circle.gsp in the Parabola folder. You’ll see a circle with center at point C that passes through point A and is tangent to a line at point B. Drag point B. Why does point C trace a parabola? 2. A parabola can be described as an ellipse with one focal point at infinity. Open the sketch Conic Connection.gsp in the Parabola folder. You’ll see the ellipse and circle from the Folded Circle construction. Press the send focal point to “infinity” button. Point A—a focal point of the ellipse and the center of the circle—will travel far off the screen. As point A moves, notice how this affects the appearance of the visible portion of the circle. Compare what you see to the Folded Rectangle construction. In what ways do they appear similar? 3. Use the illustration from your parabola proof to show that ⇧GDF = ⇧ADC. The sketch Headlights.gsp in the Parabola folder illustrates a nice consequence of this result. Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Chapter 2: Parabolas • 41 The Expanding Circle Construction In response to those who advised him to take life easy, Ibn Sina is said to have replied, “I prefer a short life with width to a narrow one with length.” He died at the age of 58. Name(s): In this activity, you’ll explore a littleknown parabola construction from the tenth century. The method originates from Ibn Sina, a jackofalltrades who was a physician, philosopher, mathematician, and astronomer! Constructing a Sketchpad Model 1. Open a new sketch. Choose Show Grid from the Graph menu. Then choose Hide Grid to remove the grid lines while keeping the x and yaxes. G D H 6 4 C 2 2. Label the origin as point A. 3. Choose the Compass tool. Click on the yaxis above the origin (point C) and then below the origin (point B). This creates a circle with center at point C passing through point B. A 5 E F 2 5 B 4. Construct point D, the intersection of the circle and the positive yaxis. 5. Construct points E and F, the intersections of the circle and the xaxis. 6. Construct lines through points E and F perpendicular to the xaxis. 7. Construct a line through point D perpendicular to the yaxis. 8. Construct points G and H, the intersections of the three newly created lines. If you don’t want your traces to fade, be sure the Fade Traces Over Time box is unchecked on the Color panel of the Preferences dialog box. 9. Select points G and H and choose Trace Intersections from the Display menu. Drag point C up and down the yaxis and observe the curve traced by points G and H. The curve you see is the locus of points G and H as point C travels along the yaxis. 10. Drag point B to a new location, but keep it below the origin. Then, if necessary, choose Erase Traces from the Display menu to erase your previous curve. Trace several new curves, each time changing the location of point B. For every new location of point B, you need to retrace your curve. Ideally, your parabola should adjust automatically as you drag point B. Sketchpad’s powerful Locus command makes this possible. 42 • Chapter 2: Parabolas Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press The Expanding Circle Construction (continued) 11. Turn tracing off for points G and H by selecting them and once again choosing Trace Intersections from the Display menu. 12. Now select points G and C. Choose Locus from the Construct menu. Do this again for points H and C. You’ll form an entire curve: the locus of points G and H. Drag point B to vary the shape of the curve. Questions Q1 As you drag point B, which features of the curve stay the same? Which features change? Q2 The creator of this technique, Ibn Sina, didn’t, of course, have Sketchpad available to him in the tenth century! How would this construction be different if you used a compass and straightedge instead? The Geometric Mean It certainly looks like the Expanding Circle method draws parabolas, but to prove why, you’ll need to know a little about geometric means. ab . The geometric mean x of two numbers, a and b, is equal to Equivalently, x2 = ab. Thus the geometric mean of 4 and 9 is (4)(9) = 6 It’s possible to determine the geometric mean of two numbers geometrically rather than algebraically. Specifically, if two segments have lengths a and b, we can construct—without measuring—a third segment of length a b . 13. Open the sketch Geometric Mean.gsp in the Parabola folder. You’ll see a circle whose diameter consists of two segments with lengths a and b laid side to side. A chord perpendicular to the diameter is split into equal segments of length x. x a b x 14. Use Sketchpad’s calculator to compute the geometric mean of lengths a and b. Compare this value to x. Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Chapter 2: Parabolas • 43 The Expanding Circle Construction (continued) Questions Q3 The second page of Geometric Mean.gsp outlines a proof showing that x is the geometric mean of a and b. Complete the proof. How to Prove It With your knowledge of geometric means, you can now prove that points G and H of the Expanding Circle construction trace a parabola. Since the location of point H changes as the circle grows and shrinks, it’s labeled below as (x, y), using variables as coordinates. To make things more concrete, we’ll assume AB = 3. G H = (x, y) D 6 4 2 C A = (0, 0) E 5 F 5 2 B (0, 3) Questions The questions that follow provide a stepbystep guided proof. You can answer them or first write your own proof without any hints. Q4 Fill in the lengths of the following segments in terms of x and y: AF = AD = Use your knowledge of geometric means to write an equation relating the lengths of AB, AF , and AD . Is this the equation of a parabola? Q6 Give an argument to explain why point G also traces a parabola. Q5 Q7 Rewrite your proof, this time making it more general. Let AB = s. Explore More 1. Open the sketch Right Angle.gsp in the Parabola folder. Angle DEB is constructed to be a right angle. Drag point E and observe the trace of point G and its reflection G . Explain why this sketch is essentially the same as the Expanding Circle construction. 44 • Chapter 2: Parabolas Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Parabola Projects The projects below extend your parabola knowledge in new directions and are ideal for inclass presentations. 1. Build your own physical model of the Sliding Ruler construction. 2. Create a custom tool that automatically builds a parabola given a directrix and a focal point. 3. In the Folded Rectangle construction, you built the crease line formed when point B is folded onto point A by constructing the perpendicular bisector of segment AB. Imagine now that Sketchpad’s Midpoint and Perpendicular Line commands are broken. How can you construct the crease line without them? The picture below offers one possibility. It’s from the seventeenthcentury mathematician Frans van Schooten and shows a rhombus EBFA with a slotted rod passing through points E and F. Open the sketch Rhombus.gsp in the Parabola folder and experiment with the model. How is this model similar to the Folded Rectangle construction? What purpose does the rhombus serve? E B A F D Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Chapter 2: Parabolas • 45 3 Hyperbolas CHAPTER OVERVIEW The shadows cast on a wall by a lamp with a cylindrical shade. The paths of comets that enter the inner solar system and then leave forever. The mirrors in many reflecting telescopes. All of these are examples of hyperbolic curves. The picture at right shows a double cone being sliced by a plane. The cross section (the “conic section”) is a hyperbola. After an introduction to the distance definition of a hyperbola in activity I, a variety of hyperbola construction techniques are presented in activities II and III. In total, you’ll find seven methods to sample. Open the multipage sketch Hyperbola Tour.gsp for a handy slideshow overview. When a plane cuts both halves of a doublecone, the result is a hyperbola. The plane need not be parallel to the doublecone’s axis. I. Introducing the Hyperbola Gold coins await the treasure seekers who successfully apply the distance definition of a hyperbola. II. The Concentric Circles Construction The distance definition of a hyperbola serves as a springboard for two hyperbola constructions: one built from concentric circles, the other from a rotating ruler. III. Hyperbola Projects Round out the chapter with more hyperbola goodies. Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Chapter 3: Hyperbolas • 49 Introducing the Hyperbola Name(s): What does it take to find buried treasure? A map, a shovel—and would you believe a hyperbola or two? Yes, with nothing more than two hyperbolas, you can track down some hidden gold. You’ll be on your way once you learn the definition of these curves. What Is a Hyperbola? Here is the geometric definition of a hyperbola: A hyperbola is the set of points P such that the difference of the distances from P to two fixed points (the foci) is constant. If you open the sketch Hyperbola.gsp in the Hyperbola folder, you’ll see a hyperbola along with its foci, F1 and F2. Every hyperbola consists of two separate branches. Point P currently sits on the left branch of the hyperbola. For an ellipse, the sum of the distances from every point on the curve to the two foci remains constant. But for a hyperbola, it is the difference of the two distances that remains constant. Of course, the difference of two numbers is either a positive or a negative number, depending on the order of subtraction. If you move point P along the left branch of the hyperbola, then drag it onto the right branch, you’ll see that the value of PF2 – PF1 switches from positive to negative. Dragging either focal point to a new location will change the value of this constant difference. In a hyperbola, it is the absolute value of PF2 – PF1 that remains constant. Every hyperbola also has two lines associated with it: the asymptotes. Press the Show Asymptotes button to view them. The two branches of the hyperbola approach these lines but never touch them. Said another way, the distance between the branches and the asymptotes approaches zero, but never reaches it. 50 • Chapter 3: Hyperbolas Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Introducing the Hyperbola (continued) ʻXʼ Marks the Spot Legend has it that the island of Keypress contains a buried treasure chest of gold. After years of searching, you find the following note: My treasure is two miles farther away from the giant boulder (point B) than the lighthouse (point L). It’s also one mile farther away from the cave (point C ) than the jail (point J). Open the sketch Treasure.gsp in the Hyperbola folder to view the landmarks in the note. Can you pinpoint the exact spot where the treasure is buried? As help, the sketch contains two hyperbolas: one with foci at B and L, the other with foci at C and J. You can change the constant difference associated with each hyperbola by dragging the segments that sit along the left edge of the sketch. Explain your method and why it works. Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Chapter 3: Hyperbolas • 51 The Concentric Circles Construction Name(s): Circles that are concentric share the same center. In this activity, you’ll use two sets of concentric circles to draw a hyperbola by hand. You’ll then transfer this technique to Sketchpad to draw a hyperbola whose shape and size can be adjusted by just dragging your mouse. Sketching Hyperbolas by Hand The illustration that follows shows two sets of concentric circles. One set of circles is centered at point F1, the other at point F2. For each set, the radii of the circles increase by 1’s, from 1 unit all the way up to 7 units. Points F1 and F2 are the foci of an infinite number of hyperbolas, but only two that we’re interested in: one hyperbola that passes through point A and another that passes through point B. Q1 How many units apart are points A and F1? How many units apart are points A and F2? What is the numerical value of AF1 – AF2? Q2 Locate and mark at least 16 points that sit on either branch of the hyperbola passing through point A. Explain how you found them. Q3 Locate and mark at least 16 points that sit on either branch of the hyperbola passing through point B. (Use a different colored pen or pencil if possible.) Explain how you found them. Q4 Using the points you found as guidelines, sketch the two hyperbolas. A B F1 52 • Chapter 3: Hyperbolas F2 Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press The Concentric Circles Construction (continued) By dropping two stones into a peaceful pond, you can create an animated version of the concentric circles. Open the sketch Ripples.gsp in the Hyperbola folder and press the Animate button to view some ripples and their accompanying hyperbolas. Examining a Sketchpad Model Now that you’ve drawn some hyperbolas by hand, it’s time to examine a dynamic one: a hyperbola that changes shape as its parts are dragged. Open the sketch Concentric Circles.gsp in the Hyperbola folder. You’ll see two circles— one red, one blue. The sizes of these two circles are controlled by the segments at the top of the screen. The red circle has radius AC and center F1, the blue circle has radius BC and center F2. A B C F1 F2 To operate this model, drag point C. As you do, the radii will adjust to remain equal to AC and BC. At the same time, you’ll be tracing the intersection points of the two circles. Drag point C to the right of point B, then back to the left of point A. Questions Q5 As you drag point C, the radii of both circles change lengths. Still, there is a relationship that exists between the two radii whenever point C is to the right of point B or to the left of point A. What is it? Q6 Explain why the intersection points of the two circles trace a hyperbola. Q7 Select your two onscreen circles, choose Trace Circles from the Display menu, then drag point C. Based on this experiment, describe the similarities between your Sketchpad construction and the concentric circles technique. Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Chapter 3: Hyperbolas • 53 The Concentric Circles Construction (continued) The Rotating Ruler Method Open the sketch Rotating Ruler.gsp in the Hyperbola folder. You’ll see the model below. The rectangle represents a ruler that rotates around the fixed point F2. A string is attached from point F1 to the corner of the ruler (point B). The string is held taut against the edge of the ruler by a pencil at point A. Pulling the pencil up along the edge of the ruler causes the ruler to rotate while point A traces a piece of a hyperbola. To operate the Sketchpad model, drag point B. F1 F2 A B How to Prove It Can you prove that the Rotating Ruler draws hyperbolas? Try developing a proof on your own or work through the following questions. Questions Q8 If the pencil at point A does indeed trace a hyperbola with foci at F1 and F2, then what value must you prove constant? Q9 Explain why the following equality holds: (BA + AF2 ) – (BA + AF1 ) = constant Q10 Complete the proof. 54 • Chapter 3: Hyperbolas Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Hyperbola Projects The projects below extend your hyperbola knowledge in new directions and are ideal for inclass presentations. 1. In the Folded Circle Construction (Chapter 1) you picked a point B within a circle and then folded the circle repeatedly so that points on its circumference landed on B. The outline of creases formed an ellipse. If you’ve saved your sketch from that activity, open it. If not, open Folded Circle Revisited.gsp in the Hyperbola folder. Hint: In the Folded Circle Construction, you connected points A and C with a segment. Try something similar here. What happens when point B sits outside the circle? Drag point B to find out. (You might want to model this construction with paper also—draw a circle on a sheet of notebook paper, mark a point B outside the circle, then fold point B repeatedly onto different points along the circumference.) Can you prove that this modified construction generates hyperbolas? To do so, you’ll need to find which point along each crease line is tangent to the hyperbola. 2. In Project 1 above, you built the crease line formed when point C is folded onto point B by constructing the perpendicular bisector of segment BC. Imagine now that Sketchpad’s Perpendicular Line and Midpoint commands are broken. How can you construct the crease line without them? The linkage below from seventeenthcentury mathematician Frans van Schooten offers one possibility. It shows a rhombus FBGC with a slotted rod passing through points F and G. What purpose does the rhombus serve? How is this model similar to that in Project 1? F C A B G E Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Chapter 3: Hyperbolas • 55 Hyperbola Projects (continued) 3. Open the sketch van Schooten.gsp in the Hyperbola folder. You’ll see a working model of the linkage below from the seventeenthcentury mathematician Frans van Schooten. Drag point C and observe the trace of point E. Can you prove that point E traces a hyperbola? The second page of the sketch contains some hints to get you started. C A B F E 4. The sketch Tangent Circles.gsp in the Hyperbola folder shows a red circle c3 that’s simultaneously tangent to circles c1 and c2. Press the Animate button and observe the path of point C, the center of circle c3. Can you prove that C traces a hyperbola? 5. Rectangular hyperbolas are of the form xy = c, where c is a constant. Open the multipage sketch Area.gsp in the Hyperbola folder to learn about some applications of these curves. 56 • Chapter 3: Hyperbolas Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press 4 Optimization CHAPTER OVERVIEW What do a burning tent, a circular swimming pool, and a cowgirl have in common? Nothing, perhaps, except that all three appear in this chapter in geometric optimization problems. Can you find a speedy path to your burning tent before only ashes remain? Can you swim to your friend with only a minimum of effort? Can you help a cowgirl lead her horse to food and water by plotting the shortest riding distance? Normally, the topic of optimization doesn’t arise until calculus, and there it is treated algebraically. Calculus is a great tool, but it’s not Ellipses aren’t generally considered standard firefighting tools, but perhaps the only way to solve optimization they should be . . . problems. There’s only one prerequisite for this chapter—the PinsandString Construction from Chapter One. With that alone, you’re ready to approach optimization problems from a purely geometric perspective. Follow the activities in the order listed—they’re sequenced to build on each other. I. The Burning Tent Problem With your camping tent on fire, it’s mathematics to the rescue as you determine the optimal location for collecting some muchneeded water. II. The Swimming Pool Problem A lazy day in a swimming pool turns mathematical when you agree to buy the next round of ice teas for a friend. Can you find the shortest distance to paddle without breaking a sweat? III. An Optimization Project Apply your optimization knowledge to this cowgirl conundrum. Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Chapter 4: Optimization • 59 The Burning Tent Problem Name(s): Consider the following situation: Ah, the great outdoors. Camping, the fresh air, the starry night sky, and—fire! Your tent is ablaze! Fortunately, you (at point Y ) have a bucket in hand. You decide to run to the river’s edge, fill your bucket, and race to your tent (point T ) to douse the flames. Where along the river should you head in order to minimize your total running distance? The picture below shows one possible location—point P—you might run to. Yes, we admit it: If our tent were really on fire, we wouldn’t stop to do math either! This is an example of an optimization problem. You’re trying to optimize the distance YP + PT to make it as small as possible. This problem can be solved algebraically, but it becomes very messy. So instead, you’ll investigate the situation geometrically, first with string, then with Sketchpad. Constructing a Physical Model Text Preparation: You’ll need a sheet of paper (larger is better), some string, a pencil, a ruler, and tape. 1. Draw a long line on your paper to represent the river’s edge. Leave space below the line to represent the water. Draw two points, Y and T, to represent you and the tent. 2. Cut a length of string and attach its ends with tape or thumbtacks to points Y and T. If the string does not extend below the river’s edge when pulled taut, cut a longer piece or relocate Y and T. Questions Q1 Without taking any measurements, use your string to find the optimal location along the river to run. Explain your method. 60 • Chapter 4: Optimization Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press The Burning Tent Problem (continued) The Burning Tent problem can be solved in several ways. The questions that follow describe one especially interesting technique. As you answer the questions, think about the similarities between your method for finding the optimal river location and the one presented here. Before you begin, make sure your string is set up as before, with its ends attached to points Y and T. The string should be long enough to extend below the river’s edge when pulled taut. Q2 Use a pencil to pull the string taut so that the pencil point sits directly on the river’s edge. Call this point P. Point P is one location along the river you might run to. It’s probably not the best location, but it will serve as an initial guess. Use your string to find another point along the river’s edge equivalent to P in total running distance. Describe how you found the point. Q3 Ignore, for a moment, the context of this problem. Use your string and a pencil to draw all points, whether on land or in the water, equivalent to P in total distance. Describe this set of points. What curve have you drawn? Q4 Use the curve from Q3 to identify two intervals on the river’s edge: those locations whose total running distance is less than that of P’s and those locations whose total running distance is greater than that of P’s. Explain how you found these intervals. Q5 How should you proceed in order to find the optimal location along the river? Constructing a Sketchpad Model The ideas behind the string solution to the Burning Tent problem can be applied to a corresponding Sketchpad model. Here’s how: You may have built such a tool if you did Project 6 from Ellipse Projects (page 29). If so, use it instead. 3. Open the sketch Burning Tent.gsp in the Optimization folder. You’ll see “you” (point Y), the tent (point T), and a point P along the river you might run to. 4. Choose Ellipse by Foci/Point from the Custom Tools menu in the Toolbox. This tool takes any three points—F1, F2, and P—and constructs an ellipse with foci at F1 and F2, passing through point P. Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Chapter 4: Optimization • 61 The Burning Tent Problem (continued) Questions Q6 Use the Ellipse by Foci/Point tool to find the optimal river location to run to. What is the relationship between the ellipse and the river at the optimal location? Q7 Can there be more than one optimal location for a straight river? Explain. Q8 Draw a curvy river (by hand, if you prefer) for which there are two optimal locations to run to. Include an ellipse in your drawing to justify your answer. Q9 Suppose that regardless of where you ran to along the river’s edge, the total distance from you (point Y) to the river to the tent (point T) was the same. What would such a river look like? A Reflection Technique The Burning Tent problem is quite old. It appears in the 1917 book Amusements in Mathematics by the great puzzlemaster Henry Ernest Dudeney, where it’s called the “Milkmaid Puzzle.” Dudeney’s method for solving the problem is totally different from the one you just used. Follow the steps below to understand his approach. 5. Open the sketch Reflection.gsp in the Optimization folder. You’ll see you (point Y), the tent (point T), and an arbitrary point P along the river you might run to. 6. Doubleclick the line representing the river’s edge to mark it as a mirror line of reflection. The line should flash briefly to indicate that it’s been marked. 7. Select point T using the Arrow tool. Then choose Reflect from the Transform menu to reflect point T across the river’s edge. Label the reflected point as T . 8. Connect point P to point T with a segment PT . Questions Q10 Explain why the following equality holds for any location of point P: YP + PT = YP + PT Q11 Based on the equality from Q10, explain how to find the optimal location of point P. Then use the ellipse tool to see if it yields the same result. Q12 Solve this problem by reflecting point Y instead of point T. 62 • Chapter 4: Optimization Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press The Swimming Pool Problem Name(s): Now that you’ve put out the flames from the Burning Tent, how about a wellearned rest? It’s just you, a friend, and a swimming pool. But wouldn’t you know it—the ellipse from the Tent problem returns! Read on . . . Tea for Two You (at point Y ) and your friend (at point F ) are floating on inflatable lounge chairs in a circular swimming pool. Waiters are positioned all around the edge of the pool, and it’s your turn to buy two iced teas. You’ll need to paddle to the edge, buy the drinks, and deliver one to your friend. Where along the pool’s edge should you paddle to in order to minimize the total distance? The picture below shows one possible path. P F (friend) Y (you) Try changing the location of points Y and F. Press the Reset button when you’re done. Open the first page of the sketch Swimming Pool.gsp in the Optimization folder. Besides the swimming pool, you’ll see an ellipse with foci at points Y and F that passes through an arbitrary point P on the circle’s circumference. Drag point P. The ellipse adjusts itself, with Y and F remaining the foci. Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Chapter 4: Optimization • 63 The Swimming Pool Problem (continued) Questions Q1 Notice that there are four positions of P where the ellipse is tangent to the circle. As with the Burning Tent, these locations are special. Use the four circles that follow to draw the four tangent ellipses. F (friend) Y (you) F (friend) Y (you) F (friend) F (friend) Y (you) Q2 Y (you) Now open the second page of the sketch and drag point P. You’ll see that four locations on the circle’s circumference—P2 through P5— correspond to the points of tangency you drew above. The sketch also includes two more points, P1 and P6, for reference. Here, in no particular order, are descriptions of locations P2 through P5: • This location gives the shortest overall paddling distance. • This location gives the longest overall paddling distance. • This location isn’t the best, but it’s better than all nearby locations on either side of it. • This location isn’t the worst, but it’s worse than all nearby locations on either side of it. Match the descriptions to the locations. Explain how the positions of the ellipse, relative to the pool, allowed you to draw your conclusions. Q3 From shortest overall paddling distance to longest, rank the four locations P2, P3, P4, and P5. 64 • Chapter 4: Optimization Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press The Swimming Pool Problem (continued) Q4 Below is a set of axes for graphing. The horizontal axis represents positions along the edge of the pool. For convenience, six locations are labeled—P1 through P6. (Imagine the pool’s edge cut and then straightened into a segment.) The vertical axis represents the total distance it takes to paddle to your friend. Use the information from Q2 and Q3 to draw a rough graph of the location along the edge versus total paddling distance. The graph need not represent the actual distances; rather, strive to make the shape of the graph and the relative heights at P1 through P6 as accurate as possible. Distance to paddle P1 P2 P3 P4 P5 P6 Location along the pool’s edge Q5 Believe it or not, you’ve just done some calculus! We use the following terms in calculus to describe points on a graph: absolute maximum, absolute minimum, relative maximum, relative minimum. Make an educated guess as to the meanings of these terms, then match the terms to points P2 through P5. Q6 If you open page 3 of Swimming Pool.gsp and press the Show Graph button, you’ll see the graph from Q4. Experiment with its shape by moving the locations of Y and F. How should Y and F be positioned to make the entire graph horizontal, or nearly so? Exploring Conic Sections with The Geometer’s Sketchpad © 2012 Key Curriculum Press Chapter 4: Optimization • 65 An Optimization Project The project below builds on your knowledge from the Burning Tent and Swimming Pool problems. The Cowgirl Problem Open the sketch Cowgirl.gsp in the Optimization folder and consider the following situation: A cowgirl wants to give her horse some food and water before returning to her tent. She starts at point C and decides to travel first to the pasture, then to the river, and then back to her tent. What path should she take to minimize her riding distance? Pasture Points A and B are two possible locations the cowgirl might take her horse to. C (Cowgirl) T (Tent) A B River Solve this problem in two ways: with ellipses (use the provided custom tool) an