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Pythagoras Plugged In: Proofs and Problems for The Geometer's Sketchpad
Pythagoras Plugged In: Proofs and Problems for The Geometer's Sketchpad
Dan Bennett
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Pythagoras Plugged In guides students through a variety of proofs and applications of the Pythagorean theorem. By constructing and dynamically manipulating figures, students visualize the theorem and gain insights that no static illustration can offer. In addition to the activities, Pythagoras Plugged In includes historical background on Pythagoras and the theorem, teacher’s notes, and an appendix on creating animated presentation sketches.
The sketches referenced in the book can be downloaded from https://sketchpad.keycurriculum.com/KeyModules/index.html.
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Year:
2003
Publisher:
Key Curriculum Press
Language:
english
Pages:
103
ISBN 10:
1559536497
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PDF, 704 KB
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Project Editor Editorial Assistant Production Editor Copyeditor Manager, Editorial Production Production Director Production Coordinator Cover Designer Cover Photo Credit Prepress and Printer Dan Ditty Shannon Miller Kristin Ferraioli Erin Milnes Deborah Cogan Diana Jean Parks Charice Silverman Kavitha Becker M. L. Sinibaldi/Corbis Data Reproductions Executive Editor Publisher Casey FitzSimons Steven Rasmussen ®The Geometer’s Sketchpad, ®Dynamic Geometry, and ®Key Curriculum Press are registered trademarks of Key Curriculum Press. ™Sketchpad is a trademark of Key Curriculum Press. All other registered trademarks and trademarks in this book are the property of their respective holders. Limited Reproduction Permission © 2003 Key Curriculum Press. All rights reserved. Key Curriculum Press grants the teacher who purchases Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad the right to reproduce activities and sample sketches for use by his or her own students. Unauthorized copying of Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad or of the Pythagoras Plugged In sketches constitutes copyright infringement and is a violation of federal law. Pythagoras Plugged In CDROM Key Curriculum Press guarantees that the Pythagoras Plugged In CDROM that accompanies this book is free of defects in materials and workmanship. A defective CDROM will be replaced free of charge if returned within 90 days of the purchase date. After 90 days, there is a $10.00 replacement fee. Key Curriculum Press 1150 65th Street Emeryville, California 94608 5105957000 editorial@keypress.com http://www.keypress.com Printed in the United States of America. 10 9 8 7 6 5 4 3 2 1 07 06 05 04 03 ISBN 1559536497 Acknowledgments I owe thanks to all the following and, I’m sure, more: Leslie Nielsen, Sarah Block, Rob Berkelman, Greer Lleuad, William Finzer, Nicholas Jackiw, Scott Steketee, Annie Fetter, Steven Chanan, Daniel Ditty, and others associated with Key Curriculum Press; Nathan King for; A FourPiece Dissection, Doris Schattschneider for Pappus’ Theorem, Daniel Scher for The Law of Cosines, Marc Roth for a new way to think about The Similar Triangle Proof, Bill Medigovich for lending me his Loomis, Michael Serra for Tessellating Pythagoras and general inspiration, David Bennett for WrongWay Squares and mathematical upbringing, Steven Rasmussen for making it possible for me to write the book; my colleague David Louis and my students at The Nueva School; and thanks especially to Leslie, Eli, and Gus. D. B. Contents Introduction .................................................................................................. vii A Little History.......................................................................................... vii What’s in This Book ................................................................................... ix To the Teacher: Using the Activities and the CDROM ........................... x Chapter 1: The Theorem .............................................................................. 1 A Right Triangle with Squares................................................................... 3 Chapter 2: Proofs and Demonstrations .................................................... 7 Square Areas on a Grid .............................................................................. 9 Pythagorean Puzzles ................................................................................. 11 The Translator Tool .................................................................................... 12 A Dissection ................................................................................................ 13 WrongWay Squares .................................................................................. 15 The Tilted Square Proof ............................................................................ 17 Behold! ........................................................................................................ 20 E. A. Coolidge’s Proof ............................................................................... 22 Ann Condit’s Proof ................................................................................... 24 Leonardo da Vinci’s Proof ........................................................................ 26 Presidential Pythagoras ............................................................................ 29 Perigal’s Proof ............................................................................................ 31 Euclid’s Proof, a.k.a. Pythagoras’ Pants .................................................. 33 A FourPiece Dissection ............................................................................ 35 Tessellating Pythagoras ............................................................................ 36 A Dynamic Proof ....................................................................................... 38 A Coordinate Proof ................................................................................... 42 The Similar Triangle Proof ....................................................................... 44 Chapter 3: Special Cases and Variations ................................................ 47 The Isosceles Right Triangle ..................................................................... 49 The 30°60°90° Triangle ............................................................................ 52 The Square Root Spiral .............................................................................. 53 A Pythagorean Tree .................................................................................. 55 Unsquare Pythagoras ................................................................................ 56 Pappus’ Theorem ....................................................................................... 58 The Law of Cosines ................................................................................... 60 Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Contents • v Chapter 4: Problems ................................................................................... 61 Activity Notes ............................................................................................... 67 Appendix: Making a Presentation Sketch .............................................. 87 References .................................................................................................... 91 vi • Contents Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Introduction The Pythagorean theorem is without question the most famous, and probably the most important, theorem in all of mathematics. This simple equation, a2 + b2 = c2, would probably be recognized by everyone who ever took a high school math class, whether they remembered what it meant or not. The legs of a right triangle are the two shorter sides. The hypotenuse is the longest side. As you may know, that equation is an abstraction of a special relationship among the three sides of any right triangle, a relationship that would probably be surprising if it weren’t so familiar. In geometric terms, the theorem states that the sum of the areas of squares constructed on the legs of a right triangle is equal to the area of the square constructed on the hypotenuse. c a b Applications of the theorem are endless—there are countless problems in which an engineer, a carpenter, a scientist, an architect, or a craftsperson might need to find an unknown or unmeasurable length, distance, height, altitude, etc. If you can set up the problem so that the unknown length is any side of a right triangle whose other two sides you do know or can measure, then you can find the length you’re looking for. Beyond practical applications of the theorem itself, the right triangle relationship pervades many branches of mathematics (most notably trigonometry), mathematics that has made the technological age in which we live possible. A Little History I’ve made some pretty lofty claims about the importance of the Pythagorean theorem. So who was this Pythagoras person whose name we associate with the theorem, and should he get all the credit? Actually, a number of different cultures around the world discovered the right triangle relationship—or at least specific cases to which it applies—independently of one another, long before Pythagoras lived. I had a history of mathematics professor who even refused to call it the Pythagorean theorem—he called it “the Babylonian theorem” in honor of the ancient civilization from which comes a stone tablet, at least 3500 years old, that lists several sets of three numbers that satisfy the Pythagorean theorem. Specific cases of the relationship have been recorded by numerous cultures—Arabic, Chinese, Indian, European, and probably others—both before and after Pythagoras’ time. Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Introduction • vii As you read this book, you’ll see that the simplicity of the Pythagorean theorem is one thing that makes it so fascinating. The theorem is such a basic means of describing the physical world that it’s even been presumed that any advanced culture would know it. In the early part of this century, when many thought there might be intelligent life on Mars or on the Moon, it was proposed that some huge illustration of the Pythagorean theorem be sculpted into Earth’s landscape. When noted mathematician Carl Friedrich Gauss was a teenager, he proposed cutting rows of trees in Siberia in the shape of a right triangle with squares on the sides. Others have suggested digging canals in the Sahara desert and burning kerosene in them to illuminate a bright sign to our Martian neighbors that we Earthlings are here and that we are reasonably smart. No one knows whether Pythagoras came upon the theorem in his studies or discovered it himself independently. By now you’re probably really wondering how such a universal theorem got its name. Pythagoras was a Greek philosopher who lived around 500 B.C.E. His claim to fame is that he proved the theorem. A proof usually offers explanation for why a theorem is true and establishes its truth for some general class of figure, rather than just specific cases. Again, just as other cultures knew specific instances of the theorem, they may also have understood how the theorem applies to any right triangle. Pythagoras himself lived and traveled throughout the Mediterranean and Asia Minor, studying and teaching in Egypt, Babylonia, and what are now Italy and Greece. Babylonia was an important center of world commerce, and during his time there Pythagoras had the opportunity to study with Babylonian, Egyptian, Chinese, and Indian scholars who may have known the theorem. One reason for the influence of Pythagoras’ school was its longevity. Pythagoras, in violation of the laws of the time, allowed and encouraged women scholars in his society. He married one of his students, and his wife and daughters kept the school active long after Pythagoras’ death. On his return to Greece, Pythagoras established a school that has had lasting influence on the study of mathematics. Pythagoras’ teacher, Thales, had begun a tradition of systematically showing how a theorem derives logically from basic principles and other theorems. Thales proved theorems about angles, including the angle sum theorem for triangles, upon which any proof of the Pythagorean theorem depends. This tradition, advanced considerably by Pythagoras’ school, is considered one of the most influential contributions to modern intellectual thought. Ironically, nobody is sure what Pythagoras’ proof of the theorem was. It may be the proof found in Euclid’s Elements (300 B.C.E.), or one of the simpler proofs you’ll study in this book. Whatever his proof, and whether or not it was original, Pythagoras’ contribution to mathematics was indisputably great, and few would begrudge him the honor of having his name attached to the theorem. Proving the Pythagorean theorem has probably interested people for as long as the relationship has been known. As you’ll see, many proofs of the theorem have a puzzlelike quality that makes them a pleasure to discover and rediscover. Renaissance artist and engineer Leonardo da Vinci came up with an original proof of the theorem, as did past United States President James Garfield. Many of the proofs in this book come from a viii • Introduction Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press book by Elisha Scott Loomis called The Pythagorean Proposition, first published in 1927, that contains more than 300 proofs! Loomis was a high school and college math teacher who made his life’s work collecting and deriving his own interesting proofs of the theorem. What’s in This Book In this book, you’ll explore the Pythagorean theorem from a number of different perspectives and strengthen your understanding of what it means and why it is true. You’ll also apply the theorem to solve problems. Along the way, you’ll learn more about the theorem’s interesting past and how it has fascinated men and women of many different cultures since the beginning of recorded history. First, you’ll be introduced to constructing right triangles and squares—the basic elements of the theorem—with The Geometer’s Sketchpad. Then you’ll do constructions that illustrate a variety of different types of proofs. In some activities, you’ll be asked to follow and supply some of the reasoning to complete the proofs. You’ll explore a variety of different types of proofs, but you may also notice common threads that run through many of them. If you experiment with enough proofs, you’ll see that there can be no limit to their number. You may even come up with an original one of your own. Once you’ve experienced some proofs of the theorem, you’ll look at special triangles, including the isosceles right triangle that may have led to another very important discovery by Pythagoras. You’ll also generalize the theorem for shapes other than squares. And you’ll explore Pappus’ theorem and the law of cosines—generalizations of the Pythagorean theorem that work even for nonright triangles. In Chapter 4: Problems, you’ll solve problems using the Pythagorean theorem, and you’ll model some of these problems with Sketchpad. Finally, there’s an optional appendix in which you can learn some of the basics of creating presentation sketches. Presentation sketches let you demonstrate your construction and explain your reasoning to a user without having to be there. Chances are you won’t do everything in this book. But I hope these activities inspire you to explore the Pythagorean theorem in depth. It’s certainly been fertile ground for many budding mathematicians, amateur and professional. Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Introduction • ix To the Teacher: Using the Activities and the CDROM The preceding introduction is written for you and your students. I encourage you to share some of the history of the Pythagorean theorem with your students, either in discussion preceding the activities or by photocopying the introduction for them. That the theorem has fascinated so many people for so many centuries may motivate your students to try to learn more about what all the fuss is about. The Activities Some of the figures students are asked to construct in this book get complicated, and all the activities assume this is not a student’s first experience with Sketchpad. Students should be introduced to the program more slowly than these activities do. The Guided Tours in the Learning Guide that comes with Sketchpad are a good place to start. Most of the activities in this book start with a short introduction followed by a section titled Sketch and Investigate. In this section, students are given the steps for constructing a figure that demonstrates the theorem. For the most part, these steps are given in geometric terms, without detailed instructions about the program’s user interface. Some of the comments in the sidebars help students find where commands are in Sketchpad’s menus. If students get stuck, make sure they look to the sidebars for hints. Interspersed with these construction steps are some questions to answer about the construction that preview ideas used in the proof. The questions are indicated with the letter “Q” preceding the question number. In some activities, the questions continue into the next section, titled Prove. Here students get down to explaining the main ideas of the proof. The place to start in this book is with the activities in the section titled The Theorem. Depending on students’ Sketchpad experience, this could take one or two class periods. It’s very important in all the activities that students be able to construct right triangles and squares. And you’ll want students to be able to use a custom tool to construct squares, too. Most of the activities in Chapter 2: Proofs and Demonstrations could probably be done in one class period each, assuming written proofs were left for homework. But unless you plan to spend a whole term on the theorem, you’re obviously not going to have students do them all. A more realistic option would be to assign groups or pairs of students different activities. Give them a class period to get started on an activity, then if possible, a longer period of time during which they can work on it outside of class. Then save a few minutes of several class periods for student demonstrations. It’s worthwhile for students to see several different proofs of the theorem, even if it’s impractical to ask them to do many. x • Introduction Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press The constructions are of varying difficulty and each activity is rated in the activity notes as beginner, intermediate, or advanced. Not coincidentally, how difficult it is to understand the proof tends to be proportional to the difficulty of the construction. It’s up to you to decide how much emphasis to give the formal proofs. The constructions and investigation questions themselves are meant to guide students to most of the insights behind the proofs, but typically there’s more explanation required for a rigorous proof. The proofs also provide opportunities to apply congruence and transformation axioms as well as, in many cases, algebra. Nevertheless, some of the harder proofs may be too much for many students. You should make sure you try the proofs yourself to determine their appropriateness for your students. In my proofs in the activity notes, I tried to err on the side of too much detail, thinking that might be safer than taking too much for granted. In general, I’d expect less detail from my students. One option to consider is to be selfish and keep the book to yourself for a year. I think you, as a math teacher, are going to get a kick out of this book. And it may serve your students better if you take time to familiarize yourself with it and try out some of the activities on your students as teacherled demonstrations. That should give you ideas about which activities would go over best with students, as well as ideas for how best to let students use the activities. The CDROM The CDROM contains sketches for Chapters 1 and 4 and folders for Chapters 2 and 3. There’s also a sketch showing various stages of the Appendix activity. Assuming your students do the constructions themselves, these sketches are only for reference. Use them to check if you or your students’ constructions came out the way they were supposed to. There’s also a custom tool for constructing a square in the Chapter 1 sketch. Again, if students make their own tool, they won’t need this one. The one exception is the activity A FourPiece Dissection. This activity requires students to start with a premade sketch. An additional folder titled Presentations contains presentation sketches—sketches with action buttons that enable users to step through dynamic, visually based demonstrations of the theorem without doing any constructions. The presentation sketches also correspond to some of the activities. They’re a bit more razzledazzle than what students construct in the activities, and they’re a means for quickly demonstrating proofs students don’t get a chance to do themselves. But they’re no substitute for the insight gained by struggling through a construction. The Chapter 4 sketch contains sketches of scale drawings and models of the problems in that chapter. Like the sketches for Chapters 1–3, these are meant to serve primarily as examples. Students should try to construct Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Introduction • xi their own models and scale drawings. There are, of course, any number of ways to model a given problem, so you needn’t be reluctant to share these examples with students to help them learn how to do their own. Be sure to read the “read me” file on the CD for an uptodate description of the CD’s contents. xii • Introduction Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press The Theorem The Pythagorean theorem, stated geometrically, says the sum of the areas of the squares on the two legs of a right triangle is equal to the area of the square on the hypotenuse. Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Chapter 1: The Theorem • 1 A Right Triangle with Squares Name(s): The basics for exploring the Pythagorean theorem are that you need to be able to construct a right triangle and squares. There are many ways to construct both these shapes. Here you’ll learn one way for each. 1. Construct segment AB. Select the segment and the point, then choose Perpendicular Line from the Construct menu. 2. Construct a line perpendicular to sAB through B. 3. Construct sAC, where point C is on the perpendicular line. This is just one of many ways to construct a right triangle. Can you come up with others? The Text tool Show or hide an object’s label by clicking on the object with the Text tool. Doubleclick on the label to edit it. The hand turns dark when positioned over an object. The letter “A” appears when it’s positioned over a label. A B C 4. Construct sBC and hide the line. Experiment with dragging each of the three vertices of the triangle. You’ll find that they behave differently because they’re constructed with different constraints. Q1 Does your triangle always stay a right triangle? What step in your construction guarantees this? Can your right triangle be every possible size or shape? Q2 Which two points behave the same? When you drag either of them, how does the triangle change? Which point or points change the shape of the triangle when dragged? 5. Relabel the sides as a, b, and c, with a the shortest leg, b the longer leg, and c the hypotenuse. Relabel the vertices opposite these sides as A, B, and C, respectively. The diagrams in this book show the right triangle positioned and labeled in the two ways shown at the top of the next page. Practice manipulating and labeling your triangle so it matches the two diagrams. Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Chapter 1: The Theorem • 3 A Right Triangle with Squares (continued) The hypotenuse in a right triangle is opposite the right angle. In this book, the hypotenuse will usually be labeled c, and the right angle vertex opposite it will be labeled C. C B C b a c a b A B c A Can you come up with other ways to construct a right triangle? The Square 6. Construct segment AB. To mark a point as center, select it and choose Mark Center from the Transform menu. The Rotate command is also in the Transform menu. B' A' 7. Mark point A as center, then rotate the segment and point B by 90°. 8. Mark point B´ as center and rotate segment AB´ and point A by 90°. 9. Connect points A´ and B to complete your square. A B 10. Select the four vertices in consecutive order and choose Quadrilateral Interior from the Construct menu. The Custom tool Making a tool of your construction will save you the work of going through the whole construction next time you want a square. 11. Select the entire figure, then press and hold the Custom tools icon. Choose Create New Tool from the Custom Tools menu. Name the new tool Square. Q3 How do different vertices behave when you drag them? Which vertices are most constrained and which are least constrained? Q4 Does your square always stay a square? What is it about your construction that guarantees your square will stay a square? 4 • Chapter 1: The Theorem Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press A Right Triangle with Squares (continued) 12. Experiment with using your new Square tool. Choose it from the Custom Tools menu. Click, move the mouse from left to right, and click again. Try to construct a second square below the first, using the same two points your tool started with, but go from right to left. It will be important for you to be able to construct a square on either side of a segment when you explore the Pythagorean theorem. B' A' B A A Right Triangle with Squares on the Sides You’re now ready to construct squares on the sides of a right triangle. If one of your squares overlaps the triangle, don’t worry: Just undo and try applying your tool to the two triangle vertices in the opposite order. A' 13. Construct a right triangle ABC, either from scratch or by using a tool. 14. Use your Square tool on the endpoints of each side of the triangle. Once you have a right triangle with squares on the sides, you have a geometric illustration of the Pythagorean theorem. 15. Measure the areas of each of the interiors and the lengths of each side. 16. Do a calculation on each side length that gives you a value equal to the area of the corresponding square. To calculate with measures, choose Calculate from the Measure menu. Click on the measures in the sketch to enter them in a calculation. B' B c a C b A 17. Perform a calculation with areas of two squares that gives you a value equal to the area of the third square. Q5 Drag different vertices of the triangle and observe the calculations. Write an equation in terms of side lengths a, b, and c that describes the relationship you observe. Q6 Does what you’ve done in this activity qualify as a proof of the Pythagorean theorem? Discuss this question with your classmates. Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Chapter 1: The Theorem • 5 A Right Triangle with Squares (continued) The Converse The converse of the Pythagorean theorem states that if the sum of the squares of two sides of a triangle is equal to the square of the third side, then the triangle is a right triangle. Follow the steps below to investigate the converse. 18. Use the Segment tool to draw an arbitrary (nonright) triangle. 19. Construct squares on the sides of this triangle. 20. Measure the areas of the three squares and calculate the sum of two of them. 21. Drag a vertex until the sum equals the area of the third square. Q7 What kind of triangle do you have when the sum of the areas of the two smaller squares is equal to the area of the larger square? Measure an angle to confirm your conjecture. 6 • Chapter 1: The Theorem Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Proofs and Demonstrations Tâbit ibn Qorra (826–901) translated Euclid’s Elements into Arabic. This is his translation of Euclid’s proof of the Pythagorean theorem. If you have trouble following this proof, remember, read from right to left. Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Chapter 2: Proofs and Demonstrations • 7 Square Areas on a Grid Name(s): In this activity, you’ll find areas of squares on a square grid, including “tilted” squares. The strategy you develop for finding these tilted square areas can be applied to one proof of the Pythagorean theorem. Sketch and Investigate 1. Open the sketch Squarefinder.gsp. 2. Drag point A or B to get a feel for how the corners of the square snap to the grid. Note that when the yellow square is tilted on the grid, a larger square with dashed sides surrounds it. Pay attention to when the measure of sAB is a whole number and when it is not. Q1 B A A perfect square is a number whose square root is a whole number. In this sketch, the yellow square has wholenumber side lengths when its sides are horizontal or vertical (that is, when the square is not tilted). In these nontilted squares, the area is a perfect square. Sketch six different perfect squares on a piece of graph paper and label each square with its side length and area. 3. Now drag point A or point B to make a tilted square whose area you think is 5 square units. When you’re pretty sure you have the right square (and not before!), press the Show Area Yellow Square button to check yourself. If you were wrong, press the Hide Area Yellow Square button and try again. The square with area 5 has a side length of the square root of 5. The measure in the sketch is a decimal approximation. Q2 Sketch the square with area 5 on your graph paper. Label it with its area and its side length expressed as a square root and a decimal approximation, that is, 5 ≈ 2.24 . Q3 Make five more tilted squares. For each, figure out the area, check yourself with the Show button, then sketch the square on your graph paper. Label each with the area and the side length. Q4 Explain your strategy for finding the areas of tilted squares. Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Chapter 2: Proofs and Demonstrations • 9 Square Areas on a Grid (continued) Explore More 1. Write an expression for the tilted square shown at right in terms of a and b. Call the side length of the tilted square c, and write an equation relating a, b, and c. Simplify the equation as much as you can. Does this equation apply even if the square is not on a grid? 2. Some tilted squares on the grid have whole number side lengths. Find as many of these as you can. b a a b b a a b 3. Is it possible to make a tilted square on the grid with area 3? Explain. 4. Is it possible to make a square on the grid whose area is not a whole number? Explain. 10 • Chapter 2: Proofs and Demonstrations Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Pythagorean Puzzles Name(s): Some demonstrations of the Pythagorean theorem are like puzzles. In these puzzles, called dissections, you start with a picture that has several pieces, which usually fit together to form one part of the Pythagorean theorem (like a2 + b2). You cut out those pieces and rearrange them to form the other part of the theorem (like c2). While simply rearranging pieces doesn’t qualify as a proof of the theorem, these puzzles can lead to proofs if you can explain why they work. Sketch and Investigate Open the sketch Pythagorean Puzzles.gsp. There are four puzzles, each on its own page. Follow the directions in the sketch. Try all four puzzles. Explore More 1. Write two expressions for the area of the square below left, one in terms of a and b and the other in terms of a, b, and c. Set the expressions equal and simplify to prove the Pythagorean theorem. b c a b a c 2. One expression for the area of the square above right is c2. Write another expression in terms of a and b. Set the expressions equal and simplify to prove the Pythagorean theorem. Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Chapter 2: Proofs and Demonstrations • 11 The Translator Tool Name(s): One common way of demonstrating the Pythagorean theorem is to cut out pieces of the squares on the legs and rearrange them to fit in the square on the hypotenuse. In this activity, you’ll create a custom tool that lets you do the same kind of thing with Sketchpad—the tool will give you pieces that you can move around freely. The tool uses a translation, which, in geometry, means a transformation that creates an identical image by sliding without turning or flipping. You’ll find this tool useful in a number of activities in this book. Sketch 1. Construct an arbitrary quadrilateral ABCD and its interior. 2. Construct a point E in a blank area. This point is where you’ll locate a copy of the quadrilateral interior in the following steps. E C B 3. Select, in order, point B and point E and choose TransformMark Vector. 4. Select only the quadrilateral interior and choose TransformTranslate and translate by the marked vector. D A 5. Drag point E to observe that this image is free to move independently of the original quadrilateral. 6. Drag any vertex of the original quadrilateral to observe that it and the translated image are dynamically linked. In the following steps, you’ll create a tool for this construction. 7. Select, in order, the original quadrilateral interior, point B, point E, and the translated interior. Press and hold the Custom tools icon and choose Create New Tool from the Custom Tools menu. Name the tool Translator. 8. Practice using the Translator tool. Here’s how it should work: Click on an interior that you wish to translate; click on one vertex of the interior; click a third time anywhere in the sketch to create a translated image with one vertex in that location. 9. Save your sketch so that you can use this tool in later activities. 12 • Chapter 2: Proofs and Demonstrations Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press A Dissection Name(s): One way to demonstrate the Pythagorean theorem is to cut the squares into pieces so that the pieces of the squares on the legs can be arranged to fit into the square on the hypotenuse (or vice versa). In this activity, you’ll learn one such dissection. Sketch and Investigate 1. Construct a right triangle and squares on the sides. How can you find the center of a square? To construct a parallel, select a point to go through and a straight object to be parallel to. 2. Construct the center of the square on side b, the larger of the two legs. B c 3. Construct a line through this center, parallel to side c, and another line perpendicular to side c. a A b C 4. Construct points of intersection of the lines with the sides of the square on side b. Select the four vertices and choose Quadrilateral Interior from the Construct menu. 5. Construct the four quadrilateral interiors of the regions into which the square on side b is divided. Give them different shades or colors. Select the lines, then choose Hide Lines from the Display menu. 6. Hide the lines. 7. Construct the interior of the square on a, the smallest leg. The four pieces of the square on side b, combined with the square on side a, make five pieces that can be rearranged to fit into the square on side c. In the following steps, you’ll make translated images of these pieces that you can drag around freely. Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press B a c b C A Chapter 2: Proofs and Demonstrations • 13 A Dissection (continued) 8. To create a translated image of the square on side a, choose the Translator tool from the Custom Tools menu, and click on the following, in order: the interior of the square, a vertex of the square, a blank area (to construct the translated copy). Repeat on the four pieces in the square on side b. B a c b C A 9. Now drag each piece into the square on side c and see if you can arrange them to perfectly fill this square. Q1 How does this demonstrate the Pythagorean theorem? 10. To confirm that this dissection works for other right triangles, change the shape of your triangle and refit the pieces. 14 • Chapter 2: Proofs and Demonstrations Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press WrongWay Squares Name(s): Typically, the Pythagorean theorem is illustrated with squares that lie outside of a right triangle. But what if you construct squares the “wrong way”? That is, what happens if one or more squares overlap the triangle? In this investigation, you’ll see how constructing overlapping squares divides your figure into pieces that can be rearranged to give a dissection demonstration of the Pythagorean theorem. Sketch and Investigate 1. Construct a right triangle. If you’re using a tool, you should be able to control which way the square goes by the order in which you drag from vertex to vertex. 2. Construct a “wrongway” square on side a so that it overlaps the triangle, and construct a square on side b that falls outside the triangle. B c a C b A 3. Construct a wrongway square on the hypotenuse, overlapping the triangle. 4. The squares on sides a and b are now divided into five regions. Construct the points where the squares intersect, then construct the polygon interiors of these regions. Q1 Drag vertices of your triangle. Does this construction always divide the square on side c into five pieces? Make note of special cases if you encounter any. 5. One of the pieces in the square on side a also lies within the square on side c, as does one of the pieces on side b. Identify the three regions that don’t already lie in the square on side c. Use the Translator tool to create translated images of these three pieces that you can drag around. Hide the original pieces and fit the translated images in the square on side c. Q2 Explain how this demonstrates that the area of the square on side c is equal to the sum of the areas of the squares on sides a and b. 6. Drag a vertex to change your triangle and refit the pieces to confirm that this works for differentshape triangles. Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Chapter 2: Proofs and Demonstrations • 15 WrongWay Squares (continued) Prove Dissection demonstrations convincingly illustrate the truth of the Pythagorean theorem, but they don’t provide any of the logic. You show that you can rearrange the pieces, but you don’t explain why this works. Why is it that you can move these pieces and they happen to fit within other places? To explain this requires a logical argument proving that each moved region is moved to a region congruent to it. Q3 Can you find three right triangles in the figure that are congruent to one another? Q4 In the labeled diagram, which three regions moved? Into which three regions did each move? B K J C A The three regions that I moved (and the three regions they moved into) are H L all right triangles. Prove that each region that moved is congruent to the region it G was moved into. Remember, to prove right triangles F E D congruent, you need only show two pairs of corresponding sides congruent (HL, LL) or one pair of sides and one corresponding pair of nonright angles (HA, LA). Explore More In this activity, you constructed two wrongway squares. Can you create a different construction that yields a dissection by constructing just one wrongway square? How about by constructing two wrongway squares different from the ones in the activity? Hint: You may need to rotate one or more pieces to fit them into the proper places when you move them. 16 • Chapter 2: Proofs and Demonstrations Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press The Tilted Square Proof Name(s): In this activity, you’ll construct a simple figure that you can use to prove the Pythagorean theorem, either by moving figures or by using a little algebra. The figure appears in very old texts and you may have even seen it in decorative tile designs. Nobody knows, but it may have been the proof known to Pythagoras, as it seems to have been known in China and India and Pythagoras studied with Chinese and Indian scholars in Babylonia. Sketch and Investigate 1. Construct a square and the center of the square. 2. Construct a point on one side of the square. Steps 1 and 2 Step 3 Steps 4 and 5 3. Mark the center of the square a center of rotation and rotate the point on the side by 90°. Repeat on this rotated image until you have points on all four sides. 4. Use the points to construct a tilted square inside the larger square. 5. Construct the interiors of the triangles in the corners of the larger square. Q1 Drag the point on the bottom side of the large square. Do the four triangles appear to remain congruent? How do you know they’re congruent? Answer in a paragraph. Q2 What kind of shape is the empty region in the center of the figure? As you change the figure, what changes about that shape? What doesn’t change? Can you make it fill the original square? How small can you make it? Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Chapter 2: Proofs and Demonstrations • 17 The Tilted Square Proof (continued) 6. Label the sides of at least one of the right triangles a, b, and c, where c is the hypotenuse. Q3 What’s the area of the tilted square inside the figure? 7. Now you’re going to move the triangles around. Use the Translator tool to construct translated images of three of the four triangle interiors. Hide the original three. 8. Arrange the triangles so that you have two rectangles as shown. Q4 What kinds of shapes are the two empty regions now? What is the sum of their areas? 9. Drag the bottom point to change the figure and reposition the triangles to demonstrate that this works for differentshape triangles. Q5 Explain how this experiment demonstrates the Pythagorean theorem. Prove You can write an algebraic proof based on the demonstration you just did. Here are some hints for planning the proof. ` Q6 Show that the tilted square inside the figure really is a square, and express its area in terms of c. Q7 What’s the length of one side of the larger square (the whole figure)? Write an expression for the area of this square in terms of a and b. Q8 What’s the area of one triangle? What’s the sum of these four areas? Q9 Write another expression for the area of the entire figure, this time in terms of the areas of the triangles and the area of the tilted square. This expression should be in terms of a, b, and c. Q10 You now have two expressions representing the same thing. Write an equation. You’re on your own from here. 18 • Chapter 2: Proofs and Demonstrations Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press The Tilted Square Proof (continued) Explore More Without too much work, you can build onto your already constructed figure to make a nice illustration of the proof. In the figure at right, the original figure was reflected over its right side. The Translator tool was used to translate triangles to existing points so that they would be attached in place, forming white squares with sides a and b, respectively. See if you can construct a figure like this that can be manipulated dynamically so that all the triangles stay congruent and the squares stay squares. By manipulating this figure, you can study many interesting special cases, such as isosceles right triangles or what happens when the triangles are made infinitely small. Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Chapter 2: Proofs and Demonstrations • 19 Behold! The figure in the Chinese text shows a 345 right triangle, but the figure works for any right triangle. Name(s): The title of this activity comes from the text of the twelfthcentury Hindu scholar, Bhaskara. In fact, “Behold!” was the only text that accompanied a figure demonstrating the Pythagorean theorem. Bhaskara must have felt the figure spoke for itself! In this activity, you’ll construct this figure. Perhaps it does speak for itself, but you can gain deeper understanding by constructing the figure and working out a proof similar to the proof outlined in the activity The Tilted Square Proof. Incidentally, this figure is also found in an ancient Chinese text, making it another candidate for being a proof known to Pythagoras. Sketch and Investigate 1. Construct a square ABCD. 2. From point D, construct a segment DE to side AB. D C 3. Construct a line parallel to sDE through point B and lines perpendicular to sDE through points A and C. 4. You should have a small, tilted square inside your original square. Construct its vertices at the points of intersection of the lines, then hide the lines and sDE. A E B 5. Construct the sides of the tilted square, then construct interiors of the right triangles surrounding it, as shown. 6. Drag point E and observe what happens to the right triangles and the tilted square. Q1 Write a paragraph about what you observe: Do E the right triangles stay right triangles? Does the square stay a square? Can you make the interior square fill the figure? If so, what kind of triangles do you get in this case? Can you make the interior square disappear? What kind of triangles will do this? 20 • Chapter 2: Proofs and Demonstrations Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Behold! (continued) 7. Using this figure for a dissection demonstration can be tricky. First, tilt the original square so that the legs of the right triangles are horizontal and vertical. The illustration below shows an outline into which the pieces can be fit to demonstrate that c2 = a2 + b2. Note: Use the Translator tool to get movable pieces, and try to form the shape below right. Your pieces may overlap the dashed line. E a b Prove This figure can be used to create an algebraic proof similar to the one you might have done in the activity The Tilted Square Proof. A possible plan follows. Read it if you want, or try it on your own. Q2 Write an expression for the area of the whole square in terms of c. Q3 Write an expression for the sum of the areas of the four right triangles in terms of a and b. Q4 The tricky part is writing an expression for the area of the tilted square. (And if you want to be very thorough, you should prove it really is a square.) The length of a side of this square can be written in terms of a and b. Write an expression for the area of the tilted square in terms of a and b. Q5 You should now be able to write an equation involving a, b, and c. You’re on your own from here. Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Chapter 2: Proofs and Demonstrations • 21 E. A. Coolidge’s Proof Name(s): This proof appears in Elisha Scott Loomis’s classic, The Pythagorean Proposition, first published by the National Council of Teachers of Mathematics in 1927. That book, now unfortunately out of print, contains over 300 proofs of the Pythagorean theorem. Many are similar to one another, and all the wellknown proofs are found in Loomis’s book. The following proof is similar to Bhaskara’s proof from the activity Behold! Loomis’s credit reads, “credited to Miss E. A. Coolidge, a blind girl. See Journal of Education, V. XXVIII, 1888, p. 17, 26th proof.” Sketch and Investigate Use a custom tool to construct these squares and the square in step 4. 1. Construct a right triangle and squares on the sides. 2. Extend sHA to point A´ by translating sHA and point A by vector HA. 3. Construct a line through point B, perpendicular to sAA´, and construct point of intersection K. 4. Construct square A´KLM. 5. Construct sBK, sGM, and sFL. 6. Hide line BK. 7. Construct the interiors of the four pieces in the square on the hypotenuse. Your figure so far may look familiar. It’s Bhaskara’s dissection from the activity Behold! F A' M E G K B L D a c C b J A H In the following steps, you’ll construct identical pieces on one of the other squares of the triangle. 22 • Chapter 2: Proofs and Demonstrations Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press E. A. Coolidge’s Proof (continued) To mark a vector, select two points, in order from tail to tip, and choose Mark Vector from the Transform menu. Be sure to choose By Marked Vector when you translate. 8. Mark vector EJ and translate the four sides and vertices of BCDE by this marked vector. 9. The region on side b is now a2 + b2. Use your Translator tool to place copies of the pieces of the square on the hypotenuse in the a2 + b2 region on side b. If you choose points carefully, you should be able to attach all the pieces to points on this region. Q1 Q2 Change your triangle. Do the corresponding pieces remain congruent? Note that leg b needs to be kept the longer leg or the construction falls apart. Describe the special case you get just before your construction falls apart. F M G B E L D A' a c C b N K A Q P J B' D' C' H Explain why the pieces can fit in the region on side b the way they do. Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Chapter 2: Proofs and Demonstrations • 23 Ann Condit’s Proof Name(s): The proof in this activity also appears in Elisha Scott Loomis’s classic, The Pythagorean Proposition. Ann Condit devised it in 1938 while a student at Central JuniorSenior High School, South Bend, Indiana. Says Loomis, “This 16yearold girl has done what no great mathematician, Indian, Greek, or modern, is ever reported to have done. It should be known as the Ann Condit Proof.” Sketch and Investigate Be sure to use point A as the second control point of your circle, rather than constructing a circle that’s “just passing through” point A. This is a different right triangle construction than you’ve done before. What guarantees the triangle is a right triangle? 1. Construct a segment AB. G 2. Construct the midpoint D of this segment. M P 3. Construct circle DA. 4. Construct sBC and sAC, where C is a point on the circle, making right triangle ABC. 5. Construct squares on the sides of right triangle ABC. F H C L E B A D 6. Construct midpoints L, M, and N of the outside sides of the squares. These three segments divide the squares on the three sides in half. Can you see why? 7. Construct segments DL, DM, and DN. K N J 8. Construct sFG, fDC, and point P where fDC intersects sFG, then hide fDC and replace it with sDP. 9. Construct the interiors of triangles DCF, DCG, and DBK. This proof relates the areas of these shaded triangles to one another and to the areas of the squares on the sides. Before you attempt the proof, you might want to investigate what these relationships are. 10. Measure the areas of the triangles and drag point C around the top half of the circle, back and forth between points B and A. Q1 How are the areas related to one another? How are they related to the areas of the squares? 24 • Chapter 2: Proofs and Demonstrations Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Ann Condit’s Proof (continued) Prove You may have discovered in the investigation that the sum of the areas of the smaller triangles is equal to the area of triangle DBK. If you can prove this is true, and if you can relate these areas to the areas of the squares, then you will have proved the Pythagorean theorem. Here are some steps for the proof. Q2 Triangles DCG, DCF, and DBK share a side length. What sides in these triangles are equal? Why? Q3 Segments PF and PG are altitudes of triangles DCF and DCG respectively. (In Loomis’s book, this is assumed. You can try to prove it. Hint: Start by showing ∆FGC ≅ ∆BAC. Then show that m∠FPC = 90° by showing that ∠CBA ≅ ∠GFC and ∠FCP ≅ ∠CAB.) The sum of these altitudes, FG, is equal to BK. Why? Q4 Show that area DCG + area DCF = area DBK. Q5 How does area DCF compare to area CFEB? Why? How do areas DCG and DBK compare to the areas of the other two squares? Q6 Now you’re on your own. Combine your answers to the questions above to write a proof of the Pythagorean theorem. Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Chapter 2: Proofs and Demonstrations • 25 Leonardo da Vinci’s Proof Name(s): Leonardo da Vinci (1452–1519) was a great Italian painter, engineer, and inventor during the Renaissance. He is famous for, among other things, painting the Mona Lisa. He is also credited with the following proof of the Pythagorean theorem. Sketch and Investigate In this figure, you don’t have to construct the square on the hypotenuse. 1. Construct a right triangle and squares on the legs. B c 2. Connect corners of the squares to construct a second right triangle congruent to the original. a A C 3. Construct a segment through the center of this figure, connecting far corners of the squares and passing through C. b D 4. Construct the midpoint, D, of this segment. The Action Button submenu is in the Edit menu. Hide Reflection 5. This segment divides the figure into mirror image halves. Select all the segments and points on one side of the center line and create a Hide/Show action button. Change its label to read Hide Reflection. 6. Press Hide Reflection. You should now see half the figure. B c a A C b D Show Reflection 26 • Chapter 2: Proofs and Demonstrations Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Leonardo da Vinci’s Proof (continued) 7. Mark D as center and rotate the entire figure (not the action button) 180° around D. 8. Select all the objects making up the rotated half of this figure and create a Hide/Show action button. Relabel the button to read Hide Rotation, but don’t hide the rotated half yet. B c a A C b D Show Reflection Hide Rotation 9. Construct xA´B and xB´A. Do you see c squared? B c 10. Construct the polygon interior of BA´B´A and of the two triangles adjacent to it. 11. Select xA´B, xB´A, and the three polygon interiors and create a Hide/Show action button. Name this Hide c squared. Going through this construction may give you a good idea of how da Vinci’s proof goes. 12. Press each of the Hide buttons, then play through the buttons in this sequence: Show Reflection, Show Rotation, Hide Reflection, Show c squared. a A C b D A' B' Show Reflection Hide Rotation Hide c squared You should see the transformation from two right triangles with squares on the legs into two identical right triangles with a square on their hypotenuses. Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Chapter 2: Proofs and Demonstrations • 27 Leonardo da Vinci’s Proof (continued) Q1 Explain to a classmate or make a presentation to the class to explain da Vinci’s proof of the Pythagorean theorem. Q2 Da Vinci’s is another of those elegant proofs where the figure tells pretty much the whole story. Write a paragraph that explains why the two hexagons have equal areas and how these equal hexagons prove the Pythagorean theorem. 28 • Chapter 2: Proofs and Demonstrations Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Presidential Pythagoras Name(s): James A. Garfield discovered a proof of the Pythagorean theorem in 1876, a few years before he became president of the United States. An interest in mathematics may not have been a prerequisite for the presidency, but it must have been common at the time. One of Garfield’s predecessors, Abraham Lincoln, credited Euclid’s Elements as being one of the books most influential to his career as a lawyer and politician, saying he learned from it how to think logically. Garfield’s Pythagorean theorem proof is illustrated with a relatively simple figure: a trapezoid. Sketch and Investigate 1. Construct a right triangle ABC and label it as shown. D A' 2. Mark point B as center and rotate side c and point A by 90°. 3. Connect points A and A´ and construct a line through point A´, parallel to side b. 4. Use the Ray tool to extend side CB and construct the point of intersection, D, of this ray and the line through A´. 5. Hide the ray and the line and replace them with segments BD and DA´. B c a C b A 6. Construct polygon interiors for the three right triangles. The point here is to use only side lengths and Calculate from the Measure menu to calculate areas. Don’t actually measure areas until you’re ready to confirm your calculations. Q1 What kind of figure is quadrilateral ACDA´? How do you know? Drag points A, B, and C. Does the figure remain this type of special quadrilateral? Q2 What can you say about the triangles into which ACDA´ is divided? Do they maintain these properties when you drag different parts of your sketch? 7. Measure sides a, b, and c. Now use just these measures to calculate the areas of the three triangles and their sum. 8. Now use the area formula for a trapezoid to calculate the area of ACDA´ using just the side lengths. (What’s the height of trapezoid ACDA´?) Construct the polygon interior of the entire figure and confirm your calculations were done correctly. Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Chapter 2: Proofs and Demonstrations • 29 Presidential Pythagoras (continued) Prove In steps 7 and 8, you calculated the area of the trapezoid in two different ways. Garfield used these two different ways of finding the area to prove the Pythagorean theorem. Can you do it too? Write two different expressions for the same area in terms of a, b, and c. Set these expressions equal and do the necessary algebra to arrive at the Pythagorean theorem. Explore More Look at the figure in the activity The Tilted Square Proof. How is Garfield’s figure related to this one? See if you can transform (that’s a hint) Garfield’s figure into the Tilted Square figure. Compare the algebra involved in the two proofs and write a paragraph about how the proofs are related. 30 • Chapter 2: Proofs and Demonstrations Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Perigal’s Proof Name(s): Many proofs of the Pythagorean theorem have ancient origins, but were rediscovered later by people unfamiliar with the older sources. This proof was “discovered” by mathematician Henry Perigal in 1873, but was probably known to the Arabian mathematician Tâbit ibn Qorra a thousand years before. Sketch and Investigate 1. Construct a square CADE. E b 2. Construct a smaller square adjacent to the original square so that the two squares share a vertex and a second vertex is attached to side AD (square AGFB in the figure). Label the sides of these squares b and a. Select points A and B in that order and choose Mark Vector from the Transform menu. Then choose Translate from the Transform menu. 3. Mark AB as a vector and translate point C by this vector. D G F a C C' A B 4. Construct sEC´ and sC´F. 5. Construct polygon interiors of the triangles. Q1 You started with two adjacent squares, and within this figure you constructed two right triangles. How are these triangles related to one another? What are the lengths of their legs? Call the length of one of the hypotenuses c. What’s the length of the other triangle’s hypotenuse? Drag point G to confirm these relationships hold for other squares. 6. Use your Translator tool to translate ∆ECC´ from point C to point G and to translate ∆C´BF from point B to point D. 7. Mark point E as center and rotate point C´ about it 90° to make square EC´FC´´. Q2 C'' E b Explain how this investigation demonstrates the Pythagorean theorem. G F a C Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press D C' A B Chapter 2: Proofs and Demonstrations • 31 Perigal’s Proof (continued) Your answer to question Q2 might serve as a good proof of the Pythagorean theorem. But it still may seem a surprise that this simple transformation from a squared and b squared into c squared always works. Here are some questions meant to guide you toward more insights into why this proof works. Q3 When you translate ∆ECC´ into ∆C´´GF, how do you know ∠ECC´ will fit into ∠C´´GF? How do you know side CC´ fits into side GF? Q4 Side CE, with length b, was translated to side GD. This new side extended above the figure by a length DC´´. How do you know this length is a? Hint: What’s the length GD? Q5 How do you know quadrilateral EC´FC´´ is a square? Q6 Combine your answers to these questions into a more complete paragraph proof of the Pythagorean theorem. 32 • Chapter 2: Proofs and Demonstrations Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Euclid’s Proof, a.k.a. Pythagoras’ Pants Russian students less reverent than Arab scholars have called the figure in Euclid’s proof Pythagoras’ pants. Name(s): The Pythagorean theorem is one of the important milestones in Euclid’s Elements. This work, written around 300 B.C.E., has had a tremendous influence on mathematics because of the systematic way in which it presents geometry propositions logically derived from one another. Euclid arrives at the Pythagorean theorem and its converse as the 47th and 48th (and final) propositions of Book 1 (out of 13). It’s thanks to Arab scholars and Moorish scholars of northern Africa and southern Spain that much of ancient Greek mathematics survived. The figure on page 7 is from a manuscript of Tâbit ibn Qorra’s translation of Euclid. Arab scholars referred to the figure as “the figure of the bride.” Sketch and Investigate 1. Construct a right triangle and squares on the sides, as shown. 2. Construct a line through the right angle vertex, perpendicular to the hypotenuse. 3. Construct sCF and sBE. D E C B A G F Euclid’s proof is pretty quick, if you can establish the relationships between the triangles and the squares. But those relationships can be subtle in the complex diagram. A little further construction can add a dynamic element to your sketch that will give you more insight into how the proof works. Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Chapter 2: Proofs and Demonstrations • 33 Euclid’s Proof (continued) 4. Construct a point K on side BC. Construct the polygon interior EAK. You should be able to drag point K back and forth along sBC. Q1 Q2 When point K coincides with point C, what can you say about the relationship between the shaded triangle and square ACDE? D E C K B Drag point K toward point B. G Does the area of the triangle change? Explain why this is so. Hint: If you consider AE as the length of the base, what’s the height of ∆EAK? Does that change? J H A F 5. Drag point K until it coincides with point B. Now mark A as center, select the triangle interior, and rotate it by 90°. Q3 What new triangle interior do you get? How are triangles EAB and CAF related? Q4 In questions Q1 and Q2, you established a relationship between ∆EAB and square EACD. How are ∆CAF and rectangle JAFH related? Prove The investigation gives you a start on Euclid’s proof. First, show ∆EAB ≅ ∆CAF. Then supply the necessary steps to show the area of square ACDE is equal to the area of rectangle AJHF. A similar argument will prove the area of the small square is equal to the area of rectangle BGHJ. 34 • Chapter 2: Proofs and Demonstrations Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press A FourPiece Dissection Name(s): Until now, all the dissection demonstrations in this book involved at least five pieces. In this activity, you’ll manipulate a premade sketch in which the squares on the legs are divided into four pieces that can be arranged to fit into the square on the hypotenuse. Investigate 1. Open the sketch 4Piece Pythagoras.gsp. In the sketch, the square on side b is divided into three pieces which, combined with the square on side a, correspond to four pieces in the square on side c. Here’s how the dissection is done: • The square on side b is divided in half. Region 1 is placed in one corner of the square on side c. 3 A 2 3 1 b c 1 a 2 C B • There will be some area left over on the side of the square on side c. This region, labeled region 2, is marked off in the other half of the square on side b. • The remaining region in the square on side b, region 3, will fit with the square on side a to fill the square on side c. Q1 To confirm this dissection works, we need to manipulate the triangle into different sizes and shapes. Start by dragging point C. Does the dissection hold up? Now drag A or B. What happened? You may have been fooled for a minute into thinking this dissection works for any right triangle. But, in fact, there is no dissection proof of the Pythagorean theorem with fewer than five pieces. Still, it seemed to hold up when you dragged C. Q2 What can you say about the triangles you formed dragging point C? Q3 You should have observed that the dissection pieces don’t stay identical when you drag point A or point B. Undo until the pieces are identical again. What’s special about this triangle? Make some measurements and calculations to see what you can discover. Can you explain why this works for this type of triangle and no other? Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Chapter 2: Proofs and Demonstrations • 35 Tessellating Pythagoras Name(s): To tessellate or tile the plane means to cover it with closed shapes that fit together without overlap or gaps. Both simple and highly decorative tessellations are common in floor and wall designs, especially in Islamic art. In this activity, you’ll create a decorative pattern that demonstrates the Pythagorean theorem. Sketch and Investigate 1. Construct a square. 2. Construct a smaller square adjacent to it, using a point on one side of the original square and one of its vertices. 3. Mark AB as vector and translate the entire figure by this vector. Repeat this translation two or three times. B A B A Select the entire figure by dragging a selection marquee around it or by choosing Select All from the Edit menu. 4. Mark CD as vector and translate the entire figure by this vector two or three times. Make sure to drag the points you started with to manipulate your tessellation. This dynamic tessellation can be quite dazzling. Once you’ve tessellated with c squared, shade or color every other c squared to get a checkerboard effect. You’ve begun to tessellate with two squares of different sizes. Note that you could continue this tessellation forever to fill an infinite plane. To demonstrate the Pythagorean theorem with the tessellation, think of these squares as a squared and b squared. It’s possible 36 • Chapter 2: Proofs and Demonstrations D B A C Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Tessellating Pythagoras (continued) to also tessellate with c squared, where a, b, and c are legs of a right triangle. It’s left to you to find points you can use for a square with side c—you actually have a number of choices. Find four points you can use to construct the interior of one square with side c and tessellate with it. Prove Write a paragraph explaining how this tessellation proves the Pythagorean theorem. Here are some questions to help guide your proof. Q1 Why is it possible to tessellate with two squares of different sizes? Q2 How do you know that the squares you constructed with side c are really squares? (Assume, as obvious, that it’s possible to tessellate by translating a single square.) Q3 For every a squared and b squared pair in the twosquare tessellation, how many c squareds are there? Explore More Experiment with different polygon interior shades, colors, and arrangements to make your tessellation look nice. Make an Animation action button that moves point B on your original square bidirectionally along the side it’s on. Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press B A Chapter 2: Proofs and Demonstrations • 37 A Dynamic Proof Name(s): Unlike dissection proofs, in which you move pieces to different locations in your figure without changing them, in this construction you’ll transform your squares, without changing their areas, to create congruent shapes. To do this, you’ll actually construct parallelograms on the sides of your right triangle. Sketch and Investigate 1. Construct a right triangle and squares on the sides. 2. Hide the far side of the square on side a and replace it with a line. In steps 3 through 5, you’ll construct a parallelogram on side a. B 3. Construct sCF, where F is a point on the line. C 4. Construct a line through B, parallel to sCF. c a G b A F 5. Construct the interior of parallelogram BCFG and hide the lines. This transformation is called a shear. A shear translates every point in a figure in a direction parallel to a given line by a distance proportional to the point’s distance from the line. Shearing a figure preserves its area. Drag point F to experiment with the parallelogram’s behavior. The key to this demonstration is that the parallelogram’s area is always the same as the square whose side it shares. Can you see why? 6. Construct a parallelogram on side b by the same method you used in steps 2 through 5 above, starting with segment CH so that H is a point that can be dragged. B c a G b A C F H 38 • Chapter 2: Proofs and Demonstrations J Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press A Dynamic Proof (continued) You should now have parallelograms on sides a and b that can be dragged back and forth by one vertex without changing their areas. 7. Construct a line through point C, perpendicular to side c. L 8. Construct sBK and sAK, where K is a point on this line. 9. Mark BL as a vector and translate sBK, point K, and sAK by this marked vector. Note that this concave hexagon consists of two parallelograms, BLK´K and AKK´M. M K B c a G K' C b A 10. Construct the polygon interior BLK´MAK. F 11. Drag point K to make sure polygon BLK´MAK behaves correctly: Point K´ should move when you drag K so that BLK´K and AKK´M remain parallelograms. H J If everything is working properly, you’re ready to manipulate your sketch to give a dynamic proof of the Pythagorean theorem. In your sketch, you may need to drag a different point, depending on how you constructed the parallelogram. 12. Drag point F to shear the parallelogram back and forth. Note that the parallelogram can fill the square. Note too that the parallelogram’s area doesn’t change. (You can measure to confirm this.) Finish the shear by dragging F to lie on the line through C and K. Q1 Explain why the area of the parallelogram doesn’t change. 13. Shear the parallelogram on the other leg of the triangle. Note that it too fills its square and that its area doesn’t change. Drag until point H is on the line. 14. Drag point K to shear the parallelograms in the square on c. Note that these parallelograms can fill the square and that the area of the polygon (the sum of the areas of the parallelograms) doesn’t change. Drag until point K coincides with point C. Q2 The figures on the next page show the sequence you might have performed. How does this demonstrate the Pythagorean theorem? Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Chapter 2: Proofs and Demonstrations • 39 A Dynamic Proof (continued) Stage 1 Stage 2 Stage 3 Stage 4 Prove The dynamic “proof” you perform by manipulating your sketch to transform the squares into two congruent figures may seem proof enough of the Pythagorean theorem. People might consider the details of the proof superfluous compared to the convincing power of the dynamic demonstration. Yet if you were showing this proof to someone, you’d probably want to explain what’s going on and why your sketch proves the theorem. And to make it a complete, or rigorous, proof, you’d want to supply the details. Start with a figure like that in stage 4 above. Add labels and write a complete proof. That is, show that each parallelogram is equal in area to a square (or part of a square) and show that the two pairs of parallelograms are congruent. 40 • Chapter 2: Proofs and Demonstrations Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press A Dynamic Proof (continued) Explore More Write a paragraph about any similarities you see between this proof and Euclid’s proof. A Presentation Sketch This sketch lends itself nicely to being made into a presentation sketch—a sketch whose actions can be driven by doubleclicking action buttons. In the figure below, point C´ is the translated image of point C by vector LB. Make the action buttons indicated. The Shear button is a Presentation button that sequences the first three Move buttons. The Return to Squares button presents the second set of three Move buttons simultaneously. Once the Presentation buttons are made, hide the separate Move buttons. L B K G Move F > C' N Move H > C' C J Move K > C Move F > J F Move H > K Move K > N C' H Shear K Return to Squares Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Chapter 2: Proofs and Demonstrations • 41 A Coordinate Proof Name(s): This unique proof involves putting a right triangle in the coordinate plane and using a property of slopes of perpendicular lines. Sketch and Investigate 1. Choose Define Coordinate System from the Graph menu and turn off the grid. Show the label of the origin point. Hide the unit point (1, 0), taking care not to move it. B 2 c D 2. Construct a point B anywhere in the first quadrant. A a b C E 2 3. Construct a line through B perpendicular to the xaxis. 4. Construct point C where this line intersects the axis. 5. Hide the line and construct segments to make ∆ABC. Label the horizontal leg b, the vertical leg a, and the hypotenuse c. 6. Measure a, b, and c. Also measure the coordinates of point B. Q1 Describe how the coordinates of point B are related to the side lengths of the triangle. 7. Construct circle AB. 8. Construct points D and E, the two points of intersection of this circle with the xaxis. 9. Measure ∠DBE. Q2 What do you notice about this angle measure? Does it change when you drag point B? Why or why not? 10. Measure the coordinates of both D and E. Q3 Describe how these coordinates are related to a side length in the triangle and why. 42 • Chapter 2: Proofs and Demonstrations Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press A Coordinate Proof (continued) 11. Construct segments BD and BE. 12. Measure these segments’ slopes. 13. Perform a calculation on one of the slopes that gives a value equal to the other slope. Q4 Use the definition of slope to write the slopes of DB and EB in terms of a, b, and c. Q5 Use what you know about the slopes of perpendicular lines to write an equation from these slopes. Simplify the equation to get the Pythagorean theorem! Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Chapter 2: Proofs and Demonstrations • 43 The Similar Triangle Proof Name(s): The Pythagorean theorem proof most commonly found in geometry books is based on similar triangles. In this activity, though, you’ll do more than simply prove the theorem. You’ll puzzle out how similar triangles can be transformed into one another, and in the process you’ll discover a surprising generalization of the Pythagorean theorem. Sketch and Investigate 1. Construct a right triangle. 2. Construct a line through the right angle vertex C perpendicular to the hypotenuse. C 3. Construct a segment CD, where D is the intersection of the perpendicular line and the hypotenuse. Then hide the line. B A D 4. Construct the polygon interior of CBD. Negative numbers give clockwise rotations. 5. Mark point D as center, select the interior of triangle BDC, and rotate it by −90°. You should have a copy of ∆BDC positioned in the right angle corner of ∆CDA. This rotated copy may make it easier to see a relationship between ∆BDC and ∆CDA. C C B Mark Ratio and Dilate are both commands found in the Transform menu. Select two segments and choose Mark Ratio. Then select the figure you want to dilate and choose Dilate. Make sure By Marked Ratio is chosen in the Dilate dialog box. D A B D A 6. Now you want to expand the rotated triangle to fill triangle CDA. To do this, select two segments whose ratio defines a scale factor that will enlarge the small triangle to fill ∆CDA. Mark this ratio, and mark point D as a center for dilation. Dilate the rotated triangle by this marked ratio. Q1 How are ∆BDC and ∆CDA related? Explain. 44 • Chapter 2: Proofs and Demonstrations Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press The Similar Triangle Proof (continued) Q2 Compare these two triangles to the whole triangle, BCA. Complete the following similarity statements: ∆ ˜∆ ˜∆ 7. Measure sides a and b and areas BDC and CDA. Calculate a2 and b2, and compare these calculations to the areas. Q3 Q4 Write an equation relating a2, b2, and the two area measurements. C a B b D c A Consider the following statements: area ∆BDC = ka2 and area ∆CDA = kb2. Explain what these statements mean and why they are true based on your observations in step 6 and question Q3. Q5 Can a similar statement be made relating the area of ∆BCA to side c? Measure and do calculations to confirm. The investigation above highlights a very important feature of similar figures. By definition, corresponding lengths in similar figures are proportional. But the ratio of corresponding areas in similar figures is equal to the square of the ratio of corresponding lengths. So the areas of the similar triangles in your construction are proportional to the squares of their corresponding hypotenuses. The hypotenuses of the three triangles are a, b, and c. The area of the triangle with hypotenuse a can be written ka2, where k is some constant of proportionality. Q6 Clearly, area ∆BDC + area ∆CDA = area BCA. Write an equation relating a, b, c, and a proportionality constant, k. You’ve written a oneline proof of the Pythagorean theorem! Explore More The proof commonly found in geometry uses a proportion of corresponding sides in the similar triangles. Let BD in the figure above equal x. Write a proportion involving a, b, c, and x and simplify it to get the Pythagorean theorem. . Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Chapter 2: Proofs and Demonstrations • 45 Special Cases and Variations Renaissance artist Leonardo da Vinci proved the Pythagorean theorem and may have also known of this corollary: “The sum of the Mona Lisas on the legs of a right triangle is equal to the Mona Lisa on the hypotenuse.” Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Chapter 3: Special Cases and Variations • 47 The Isosceles Right Triangle Name(s): The isosceles right triangle played an important role in another great achievement of the Pythagoreans: the discovery of irrational numbers. An irrational number is a number that cannot be written as a simple ratio, p/q, where p and q are integers. This discovery was something of a blow to Pythagorean philosophy, which was based on the belief that whole numbers and ratios of whole numbers described the essence of the universe. In this activity, you’ll discover what was probably the firstdiscovered irrational number—the length of the diagonal of a unit square. Sketch and Investigate 1. Choose Preferences from the Edit menu. Set the Distance Unit to inches and the Distance Precision to hundredthousandths. 2. Construct a point A. By default, Polar should be chosen in the Translate dialog box. Point A´ is the translated image. 3. Use the Translate command from the Transform menu to translate it by 1 inch with a 0° angle. A A' 4. Use these two points to construct a square. 5. Construct a diagonal of the square. Q1 Explain why the diagonal divides your square into two isosceles right triangles. Q2 Use the Pythagorean theorem to show why the length of this diagonal is 2 inches. Now measure the diagonal in your sketch to get an approximation of 2 to the nearest hundredthousandth. If you have a calculator handy, use the square root key to confirm the precision of Sketchpad’s measurement. Copy and complete this statement on your paper. 2 ≈ 6. Use Sketchpad’s calculator to multiply the length of the diagonal by 100,000. This allows you to see more decimal places in Q3 2. Rational numbers have the characteristic that their decimal parts (if they have one) always either end or repeat. At least at the precision you see, does it appear that the decimal part of Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press 2 ends or repeats? Chapter 3: Special Cases and Variations • 49 The Isosceles Right Triangle (continued) A geometric consequence of irrational numbers is that they give lengths that are incommensurable. A simplified definition of incommensurable lengths is that they are lengths that cannot be measured exactly with the same ruler. This may have been what troubled the Pythagoreans the most. To understand what this means, suppose you started with a square whose side measured 1 meter. You could measure its diagonal with a ruler divided into tenths, hundredths, thousandths, and so on, but because the 2 1.4 1 1.5 2 2 1.41 1.42 2 1.414 1.415 decimal part of 2 never ends, you’d never get a mark that lined up perfectly with the diagonal length. Some stories (probably untrue) have it that the Pythagoreans were so upset by the discovery of irrational numbers that they banished or killed the discoverer. Can you see why that might have bugged the Pythagoreans, who believed numbers described everything? What’s perhaps most mindboggling is that a number that can’t be measured as a fraction of a whole number unit can be so easily constructed! It’s just the diagonal of a unit square. In fact, you can construct the mark for 2 on your ruler if you want. You just can’t get to it by dividing the units on your ruler into fractions. Prove The Pythagoreans probably proved that 2 was irrational using an indirect proof like the one outlined below. See if you can follow the outline and write a proof on your own paper, filling in where the outline leaves question marks. Assume 2 is rational and can be written as the ratio of two whole numbers, reduced to its lowest terms: 2 = p q (Any fraction can be reduced to its lowest terms, and the assumption that p/q is reduced means that p and q have no common divisors.) Square both sides of the equation, then multiply by q2 to get p2 = (?) q2 50 • Chapter 3: Special Cases and Variations Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press The Isosceles Right Triangle (continued) How do you know that if a square number is even its square root is also even? Think about the prime factorization of an even, square number. Multiply by q2 to get p2 = (?) So p2 is an even number, and thus so is p. Substitute 2r for p. That gives (2r)2 = 2q2, or (?)r2 = 2q2 q2 = (?) So q is also an even number, leading to a contradiction to our assumption that (?). Explore More 1. Hide half the square you made in the first part of this activity so you have an isosceles right triangle with 1inch legs. Construct squares on the sides of this triangle. What do you know about the area of the square on the hypotenuse? Does this give you an idea of where the term square root comes from? 2. The figure you constructed in the first part of this activity isn’t very dynamic. You can’t change its size or position it any way but horizontally. Start in a new sketch by constructing any line segment, making it a leg of a more general isosceles right triangle. Measure the hypotenuse and a leg and calculate the ratio of these lengths. How are the lengths of the hypotenuse and a leg of any isosceles right triangle related? Describe a shortcut for finding the length of the hypotenuse of an isosceles right triangle if you know the length of a leg. Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Chapter 3: Special Cases and Variations • 51 The 30°60°90° Triangle Name(s): The 30°60°90° triangle, half an equilateral triangle, has properties that make it easy to find the other side lengths if you know one side length. This construction takes advantage of the fact that an equilateral triangle is also equiangular. Can you find other methods for constructing an equilateral triangle? Sketch and Investigate 1. Construct segment AB. B' 2. Mark point A as center and rotate the segment and point B by 60°. 3. Connect points B´ and B. 4. Construct point C, the midpoint of sAB, and construct xB´C. Q1 You constructed segment B´C to be a median, but what else can you say about that segment? What are the measures of angles ACB and AB´C? A C 5. Hide the right half of your triangle and supply the new segments necessary to make 30°60°90° triangle ACB´. B B' 6. Measure lengths and ratios, if necessary, to look for relationships between AC and AB´ and between AC and B´C. Q2 Q3 Let AC = a and write an expression for AB´ in terms of a. Use the Pythagorean theorem to write an expression for CB´ in terms of a. Use Sketchpad to calculate hundredthousandth: A C 3 to the nearest 3 ≈ Explore More 1. Prove that 3 is irrational by indirect proof. First, suppose it is rational and can be written as p/q, where p and q are integers with no common divisors. Then show that if p/q = a common divisor of 3. 3 , then p and q must have 2. Construct all three medians in your equilateral triangle. What kinds of triangles are formed? Call the shortest segment in your figure x, then express as many lengths as you can in terms of x. 52 • Chapter 3: Special Cases and Variations Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press The Square Root Spiral Name(s): Irrational numbers like 2 and 3 correspond to points on a ruler, but you could never find those points precisely by dividing your ruler into fractional parts. Interestingly, you can construct square roots with compass and straightedge (or with Sketchpad). In this activity, you’ll construct a square root spiral and a table of approximate square roots. Sketch and Investigate The first part of the spiral is a right triangle with one 1inch leg. Make sure to choose inches for your Distance Unit under Preferences in the Edit menu. 1. Construct a segment AB. 2. Translate point B by 1 inch and 0° to construct point B´. A 3. Construct circle BB´. 4. Construct a line perpendicular to sAB. Drag, if necessary, so that the line does not appear to go through point B´. C B B' 5. Construct point C at the intersection of the circle and the perpendicular line. 6. Hide the circle, point B´, and the line. 7. Construct sBC and sAC. 8. Measure sAC. So far you have a right triangle in which one leg can be any length and the other is always 1 inch long. Next, you’ll iterate this construction to create a spiral of right triangles with 1inch legs. Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Chapter 3: Special Cases and Variations • 53 The Square Root Spiral (continued) 9. Select points A and B and choose Iterate from the Transform menu. 10. Map point A to itself and point B to point C, then click Iterate. 11. With the iterated image selected, use the + and – keys on the keyboard to increase or decrease the number of iterations (and thus the number of triangles and entries in the table). 12. Drag point A or point B and observe the effect on the spiral. This changeable spiral is fun to manipulate, but we want to look at a specific spiral—one in which the first spiral arm has a length of 1 inch. D 13. Construct a point D anywhere in the sketch and translate it by 1 inch and –90° to make a point D´ directly below it. D' C 14. Merge point A to point D, then merge point B to point D´. Q1 Study the table generated by the iteration. The second entry in the table (n = 1), corresponds to the second spiral arm—the hypotenuse in the original construction. What is this length in radical form? Q2 What does the third entry in the table (n = 2) represent? Use the Pythagorean theorem to find this number in radical form. Q3 Make a table on your paper of square roots and their decimal approximations for the numbers 1–16. Q4 Explain why this construction generates square roots of consecutive whole numbers. 54 • Chapter 3: Special Cases and Variations Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press A Pythagorean Tree Name(s): With Sketchpad’s iteration capabilities, you can construct part of the Pythagorean theorem figure (a square with a right triangle on one side), and then iterate the construction as many times as you like to create a fractallike Pythagorean tree. The tree isn’t likely to provide any new insight into the theorem, but it’s fun to look at and play with! Sketch and Investigate 1. Construct a square ABCD that’s controlled by the two bottom points A and B. 2. Construct the midpoint E of the top side CD and construct circle EC. F E D C A B 3. Select, in order, point C, point D, and the circle. Choose ConstructArc on Circle. 4. Hide the circle and construct point F on the arc. 5. Construct sides CF and DF. Steps 6 and 7 will add some color “special effects” to your sketch when you’re done. 6. Measure one side of the square. 7. Select this measurement and the square interior and choose DisplayColorParametric. Enter a range from 0 to about what the current side length is. 8. Select points A and B and choose TransformIterate. Map points A and B to points A and F. In the Structure popup menu, choose Add New Map and map points A and B to points F and B. Also in the Structure popup menu, uncheck Tabulate Iterated Values. Click Iterate. You can add branches by selecting any part of the iterated image and pressing + or – on your keyboard. You should now have squares and right triangles growing off the legs of your original right triangle. The colors of the squares are determined by their side lengths. Drag point F to wave the tree’s branches and watch these colors change. Q1 What’s the relationship between any triangle in the figure and the two triangles in the next iteration? Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Chapter 3: Special Cases and Variations • 55 Unsquare Pythagoras Name(s): In the activity The Similar Triangle Proof, you arrived at the Pythagorean theorem by way of an even more general statement, stated algebraically as ka2 + kb2 = kc2, where k is a constant. Whereas the algebraic terms a2, b2, and c2 can represent the areas of squares with side lengths a, b, and c, the terms ka2, kb2, and kc2 can represent the areas of any three similar figures whose areas are proportional to a2, b2, and c2. In this activity, you’ll use transformations to construct similar figures on the sides of your right triangle. Sketch and Investigate 1. Construct an arbitrary triangle ABC (not constrained to be a right triangle) with longest side c and shortest side a. C' 2. Construct an arbitrary pentagon (or any polygon) and its interior on side BC. C 4. Mark angle C´CA as an angle for rotation and mark point C as center. Rotate the translated interior. b a 3. Translate the interior and point C by vector BC. c A B You now have a pentagon on side b identical to the pentagon on side a. Now you just need to stretch it to the right size. 5. Select side b and side a and choose Mark Segment Ratio from the Transform menu. C 6. Select the polygon interior on side b and choose Dilate from the Transform menu. 7. Hide the two intermediate interiors: the translated one and the smaller interior on side b. a b c B A Rather than going through that whole process again to get a polygon on side c, you can make a tool to do it quickly. 56 • Chapter 3: Special Cases and Variations Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Unsquare Pythagoras (continued) 8. Select, in order, the polygon on side a, the polygon on side b, point B, point C, and point A. Choose Create Tool from the Custom Tools menu. Name the tool, if you wish. 9. Choose the tool and use it by clicking on, in order, the polygon on side b, point C, point A, and point B. If your tool works correctly, it should construct a similar polygon on side c as you drag from point A to point B. 10. Now to make the triangle a right triangle, construct the midpoint D of side c and construct circle DA. Merge point C to the circle. 11. Measure the areas of the polygons, and calculate the sum of the areas of the polygons on sides a and b. Q1 What do you observe about the sum of the areas? If you can, explain why this is so. If you’re unsure, continue to question Q2. Q2 Calculate the ratio of each area to its corresponding side length a, b, or c. You’ll notice that these ratios are not equal. Experiment with editing the calculations until you get equal ratios. Q3 The ratio you found in question Q2 is the proportionality constant k in the generalized Pythagorean equation ka2 + kb2 = kc2. Explain what this equation means in relation to the polygons on the sides of the right triangle. Q4 The proportionality constant k will be the same for any three similar shapes on the sides of a right triangle. But a different set of three shapes would have a different k value. What would the value of k be for semicircles on the sides of a right triangle? Q5 What would the value of k be for equilateral triangles on the sides of a right triangle? Q6 What is the value of k for squares on the sides of a right triangle? Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Chapter 3: Special Cases and Variations • 57 Pappus’ Theorem Name(s): Pappus was a Greek geometer who lived in the third century C.E., contributing original geometry propositions at a time when the golden age of the Greek mathematicians Pythagoras, Euclid, Archimedes, and Appolonius was beginning to fade. His theorem describes a relationship that holds for areas on the sides of any triangle, not just right triangles. The Pythagorean theorem, in fact, is just a special case of Pappus’ theorem. Sketch and Investigate 1. Construct any triangle ABC. 2. Construct parallelograms of any size on two sides. D 3. Construct rays to extend the sides of the parallelograms and construct their point of intersection, D, as shown. 4. Construct ray DB and point E where the ray intersects sAC. B A E C To translate by a vector you first need to select the vector endpoints and choose Mark Vector from the Transform menu. Then choose Translate. 5. Translate point E by vector DB. If you did the activity A Dynamic Proof, this figure should seem familiar. 8. Measure the areas of the parallelograms and look for a relationship among these areas. 6. Construct a line through E´, parallel to sAC. G E' F 7. Construct parallelogram ACFG, where sFG is on the line through point E´. Also construct the interior. Q1 Q2 How are the parallelograms related? Drag different points to change your triangle and the first two parallelograms you constructed. Does the relationship hold for any triangle? For any parallelograms constructed on two sides? Drag point F (or point G, if point F doesn’t cooperate) so that sCF and sAG are parallel to sEE´. Each of the parallelograms on sides AB and BC are equal to part of parallelogram ACFG when it’s aligned this way. Which parts are equal and why? 58 • Chapter 3: Special Cases and Variations Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Pappus’ Theorem (continued) Q3 Manipulate the first two parallelograms so that they approximate squares. Is the third parallelogram ever a square? Under what conditions? Prove In the Sketch and Investigate section, you discovered Pappus’ theorem that the sum of the areas of two parallelograms on two sides of any triangle is equal to the area of a specially constructed parallelogram on the third side. If you were able to answer question Q2, you discovered that the third parallelogram could be manipulated and then divided into two parallelograms, each equal in area to one of the first two parallelograms. To prove a pair of parallelograms have equal area, show that each has area equal to a third parallelogram. Points K and L in the diagram at right should give you a hint. Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press D L K B A E G C E' F Chapter 3: Special Cases and Variations • 59 The Law of Cosines Name(s): The Pythagorean theorem gives you a way to find the length of one side of a right triangle if you know the lengths of the other two sides. Trigonometry builds on the Pythagorean theorem with theorems that enable you to find the measures of all the parts of a triangle given measures of parts that determine the triangle. With the law of cosines, you can find the measure of an unknown side of any triangle given the measures of the other two sides and the angle between them (SAS). In this activity, your construction will demonstrate the law of cosines for obtuse triangles. To understand how the figure demonstrates it, you’ll need to know a little trigonometry. Sketch and Investigate 1. Construct a square and label the bottom side c. Any reference to ∠C in this activity will refer to ∠ACB. 2. Construct ∆ABC where C is any point inside the square that makes ∠C obtuse. Give segments AC and BC the labels b and a. G H D F E K 3. Construct a square on side b. Make sure to use Parallel Line from the Construct menu. Don’t just “eyeball” it. 4. On side a of the triangle, construct parallelogram CBEF as shown. C a b A c B 5. On the side of this parallelogram opposite side a, construct another square. 6. Construct segments GH and HK to complete a second parallelogram, as shown. Construct the interiors of the parallelograms and squares. Q1 Drag point C. What do you notice about the two parallelograms (as long as ∠C is obtuse)? What happens when ∠C becomes a right angle? Q2 The area of your original square is c2. How does that compare to the sum of the four areas you constructed (the two squares and two parallelograms)? Q3 If you know some trigonometry, write an expression for the area of one of the parallelograms in terms of a, b, and ∠C. Q4 Use this expression and what you discovered about the sum of the areas of the squares and parallelograms to complete the law of cosines: c2 = (?) 60 • Chapter 3: Special Cases and Variations Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Problems Even without reading this problem, you can probably guess what it’s about. This problem is from a work by Yang Hui (1261 C.E.) of China. See problem 6 on page 64 for a translation. Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press Chapter 4: Problems • 61 Problems Name(s): Right triangles can be used to model all kinds of realworld problems. That’s why the Pythagorean theorem shows up early in the history of so many cultures’ mathematics. Use the Pythagorean theorem to solve the following problems. Once you’ve solved them, model selected problems from the first 12 or make scale drawings with Sketchpad and take measurements to confirm your answers. 1. A rope from the top of a sailboat mast is fastened to the deck 4 meters from the base of the mast. If the rope is 10 meters long, how tall is the mast? 10 m mast 4m 2. I drive at 40 mph from my home to work. I have to drive 8 blocks south, then 6 blocks west. (The blocks are all the same length.) My faithful pet pigeon decides to follow me to work. He can fly straight there at 30 mph. Who will arrive at the office first, me or my bird? home 8 bl office 3. A baseball diamond is a square with 90ft sides. The pitcher’s mound lies on the diagonal between home plate and second base, 60 ft 6 in. from home plate. Is the mound in the center of the diamond? If not, how far off is it? 3rd 2nd mound home Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad © 2003 Key Curriculum Press 6 bl 90 ft 1st Chapter 4: Problems • 63 Problems (continued) 4. How long a board should you cut to make a diagonal brace for a door 2 m high and 1 m wide? 2m 1m 5. This problem is from the Chinese manuscript Arithmetic in Nine Sections (1115 C.E.): “There grows in the middle of a circular pond 10 feet in diameter a reed which projects 1 foot out of the water. W