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The Pythagorean Theorem CHAPTER The Pythagorean 9 Theorem CHAPTER 9 OBJECTIVES ● Understand the Pythagorean Theorem more deeply ● Discover the Converse of the Pythagorean Theorem ● Practice working with radical expressions ● Discover relationships among the lengths of the sides of a 45°-45°-90° triangle and among the lengths of the sides of a 30°-60°-90° triangle ● Apply the Pythagorean Theorem and its converse ● Discover and apply the Pythagorean relationship on a coordinate plane (the distance formula) ● Derive the equation of a But serving up an action, suggesting OBJECTIVES circle from the distance the dynamic in the static, has become a In this chapter you will formula hobby of mine . .The “flowing” on that ● discover the Pythagorean motionless plane holds my attention to such a ● Practice using geometry Theorem, one of the most degree that my preference is to try and make it into important concepts in tools a cycle. mathematics ● ● Develop reading M. C. ESCHER use the Pythagorean Theorem to calculate the comprehension, Waterfall, M. C. Escher, 1961 distance between any problem-solving skills, ©2002 Cordon Art B. V.–Baarn–Holland. All rights reserved. two points and cooperative behavior ● use conjectures related to the Pythagorean Theorem ● Learn new vocabulary to solve problems Escher has cleverly used right angles to form his [Ask] “What impossible things do you see?” artwork known as Water fall. The picture contains [Water seems to be traveling up an incline, yet it is three uses of the impossible tribar created by running a mill wheel.] “Which surfaces appear to British mathematician Roger Penrose (b 1931) in be horizontal? Vertical? Sloped? There are three 1954. In 1934 Swedish artist Oscar Reutersvard impossible tribars in the picture; where are they?” (b 1915), “father of impossible figures,”had created [They all have flowing water along two sides; twice an impossible tribar that consisted of a triangular one of the bars is replaced by the waterfall, and Penrose tribar arrangement of cubes. once one bar is replaced by a group of four columns.] The shapes topping the towers in Escher’s work are, on the left, a compound of three cubes and, on the right, a stellation of the rhombic dodecahedron. CHAPTER 9 The Pythagorean Theorem 461 LESSON LESSON The Theorem of 9.1 Pythagoras 9.1 In a right triangle, the side opposite the right angle is called the In a right triangle, the side opposite the hypotenuse, here with length c. PLANNING right angle is called the hypotenuse. I am not young enough to The other two sides are called legs. In c a the figure at right, a and b represent the The other two sides know everything. are legs, here with LESSON OUTLINE lengths of the legs, and c represents the b OSCAR WILDE lengths a and b. One day: length of the hypotenuse. 15 min Investigation 5 min Sharing 10 min Examples 15 min Closing and Exercises MATERIALS ⅷ construction tools FUNKY WINKERBEAN by Batiuk. Reprinted with special permission of North America Syndicate. ⅷ scissors There is a special relationship between the lengths of the legs and the length of the ⅷ Pythagorean Theorem (W) for One step hypotenuse. This relationship is known today as the Pythagorean Theorem. ⅷ Dissection of Squares (W), optional ⅷ Sketchpad demonstration Three Triangles, optional Investigation The Three Sides of a Right Triangle TEACHING You will need The puzzle in this investigation is intended to help you ● scissors recall the Pythagorean Theorem. It uses a dissection, ● a compass which means you will cut apart one or more Many students may already know ● a straightedge geometric figures and make the pieces fit into O the Pythagorean Theorem as ● patty paper another figure. 2 ϩ 2 ϭ 2 k a b c . In this lesson they Step 1 Construct a scalene right triangle in review what the letters stand for b the middle of your paper. Label the a and discover proofs showing why hypotenuse c and the legs a and b. the relationship holds for all Construct a square on each side of c right triangles. the triangle. j Step 2 To locate the center of the square on the INTRODUCTION longer leg, draw its diagonals. Label the Direct students’ attention to center O. Improving Your Visual Thinking Step 3 Through point O, construct line j Skills on page 454. Ask what they perpendicular to the hypotenuse and line k can conclude about right trian- perpendicular to line j. Line k is parallel to the hypotenuse. Lines j and k divide the square on the longer leg into four parts. gles, and help them state the Pythagorean Theorem using Step 4 Cut out the square on the shorter leg and the four parts of the square on the areas of squares and the terms longer leg. Arrange them to exactly cover the square on the hypotenuse. hypotenuse and legs. Guiding the Investigation trying a special case first—here, an isosceles right with compass and straightedge.It is also easy to triangle. As needed, point out that good pieces might create several examples using geometry software. be formed if they draw lines through the smaller One step Hand out a copy of the Step 2 As needed, remind students that the legs are squares parallel to edges of the largest square. Pythagorean Theorem worksheet the sides other than the hypotenuse, so the “longer to each group. Challenge students [Language] A dissection is the result of separating leg” is not the hypotenuse. Suggest that students to cut up one or both of the something into pieces. minimize clutter by making these diagonals very smaller squares and assemble the light or by drawing only the portion near the center Step 1 Using the Dissection of Squares worksheets or pieces on top of the largest of the square. square. As you circulate, you the Sketchpad demonstration will speed the investiga- might remind students of the tion, but the use of many different triangles drawn by Step 4 Ask students to take care in drawing and problem-solving technique of the students strengthens the inductive conclusion. cutting out pieces so they will fit together well. The constructions are quicker with patty paper than Students may want to tape the pieces together. 462 CHAPTER 9 The Pythagorean Theorem Step 5 State the Pythagorean Theorem. SHARING IDEAS You might make a transparency C-82 The Pythagorean Theorem of the Dissection of Squares In a right triangle, the sum of the squares of the lengths of the legs equals the worksheets for students to use square of the length of the hypotenuse. If a and b are the lengths of the legs, in presenting their ideas. and c is the length of the hypotenuse, then ᎏ? . a2 ϩ b2 ϭ c2 Ask about symmetry in the dissected square on the hypot- enuse. The method of the Inves- tigation gives 4-fold rotational History symmetry. [Ask] “What if the triangle isn’t Pythagoras of Samos (ca. 569–475 B.C.E.), a right triangle? Do you think depicted in this statue, is often described as “the first pure mathematician.”Samos was a there’s still a relationship among principal commercial center of Greece and is the lengths of the sides?” You need located on the island of Samos in the Aegean not answer this question now; Sea. The ancient town of Samos now lies in it’s addressed later. [Link] The ruins, as shown in the photo at right. Pythagorean Theorem is a special Mysteriously, none of Pythagoras’s writings still exist, and we know very little about his life. He case of the Law of Cosines founded a mathematical society in Croton, in ΂c2 ϭ a2 ϩ b2 Ϫ 2ab cos C, what is now Italy, whose members discovered irrational numbers and the five where ЄC is the angle opposite regular solids. They proved what is now called the Pythagorean Theorem, side c΃;when mЄC ϭ 90°, we although it was discovered and used 1000 years earlier by the Chinese and ϭ Babylonians. Some math historians believe that the ancient Egyptians also have cos C 0. used a special case of this property to construct right angles. [Ask] “What is the longest side of a right triangle?”“Is it the same as A theorem is a conjecture that has been proved. Demonstrations like the one in the longest leg?” [The hypotenuse, the investigation are the first step toward proving the Pythagorean Theorem. not the longest leg, is the longest Believe it or not, there are more than 200 proofs of the Pythagorean Theorem. side.] If you ask why the longest Elisha Scott Loomis’s Pythagorean Proposition, first published in 1927, contains side is always the hypotenuse and original proofs by Pythagoras, Euclid, and even Leonardo da Vinci and U. S. what can be said about the longer President James Garfield. One well-known proof of the Pythagorean Theorem of the two legs, you can review is included below. You will complete another proof as an exercise. the Triangle Inequality Conjecture and the Side-Angle Inequality ba Paragraph Proof: The Pythagorean Theorem b Conjecture. c 2 ϩ 2 2 a c You need to show that a b equals c for the right triangles in the figure at left. ΂ ϩ ΃2 2 ϩ ϩ 2 [Ask] “What is a theorem?” [It’s a c The area of the entire square is a b or a 2ab b . The area of any triangle 1 c a is ΂ᎏᎏ΃ab, so the sum of the areas of the four triangles is 2ab. The area of the conjecture that has been proved b 2 quadrilateral in the center is ΂a2 ϩ 2ab ϩ b2΃ Ϫ 2ab, or a2 ϩ b2.
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