CHAPTER 9 The Pythagorean Theorem

1 CHAPTER9 But serving up an action, suggestingthe dynamic in the static, has become ahobby of mine .. The flowing on thatmotionless plane holds my attention to such adegree that my preference is to try and make it into a C. ESCHERW aterfall,M. C. Escher, 1961 2002 Cordon Art B. V. Baarn rights PythagoreanTheoremOBJECTIVESIn this CHAPTER you will discover the PythagoreanTheorem, one of the mostimportant concepts inmathematics use the PythagoreanTheorem to calculate thedistance between any two points use conjectures related tothe Pythagorean Theoremto solve problemsCHAPTER 9 The Pythagorean Theorem461 Escher has cleverly used right angles to form hisartwork known as Wa t e r f a l l .The picture containsthree uses of the impossible tribar created byBritish mathematician Roger Penrose (b 1931) in1954.

2 In 1934 Swedish artist Oscar Reutersvard(b 1915), father of impossible figures, had createdan impossible tribar that consisted of a triangulararrangement of shapes topping the towers in Escher s work are,on the left, a compound of three cubes and, on theright, a stellation of the rhombic dodecahedron.[Ask] What impossible things do you see? [Water seems to be traveling up an incline, yet it isrunning a mill wheel.] Which surfaces appear tobe horizontal? Vertical? Sloped? There are threeimpossible tribars in the picture; where are they? [They all have flowing water along two sides; twiceone of the bars is replaced by the waterfall, andonce one bar is replaced by a group of fourcolumns.] CHAPTER 9 OBJECTIVES Underst and thePythagorean Theoremmore deeply Discover the Converse ofthe Pythagorean Theorem Practice working withradical expressions Discover relationshipsamong the lengths of thesides of a 45 -45 -90 triangle and among thelengths of the sides of a30 -60 -90 triangle Apply the Py thagoreanTheorem and itsconverse Discover and apply thePythagorean relationshipon a coordinate plane(the distance formula) Derive the equation of acircle from the distanceformula Practice using geometrytools Develop readingcomprehension,problem-solving skills,and cooperative behavior Learn new vocabularyPenrose 9 The Pythagorean TheoremThe puzzle in this investigation is intended to help yourecall the Pythagorean Theorem .

3 It uses a dissection,which means you will cut apart one or more geometric figures and make the pieces fit intoanother 1 Construct a scalene right triangle in the middle of your paper. Label thehypotenuse cand the legs aand a square on each side ofthe 2To locate the center of the square on thelonger leg, draw its diagonals. Label the center 3 Through point O,construct line jperpendicular to the hypotenuse and line kperpendicular to line kis parallel to the hypotenuse. Lines jand kdividethe square on the longer leg into four 4 Cut out the square on the shorter leg and the four parts of the square on thelonger leg. Arrange them to exactly cover the square on the Theorem ofPythagorasIn a right triangle, the side opposite theright angle is called the hy other two sides are called figure at right,aand brepresent thelengths of the legs, and crepresents thelength of the is a special relationship between the lengths of the legs and the length of thehypotenuse.

4 This relationship is known today as the Pythagorean am not young enough toknow Three Sides of a Right TriangleYou will need scissors a compass a straightedge patty paperOjkabcFUNKY WINKERBEAN by Batiuk. Reprinted with special permission of North America a right triangle, the side opposite the right angle is called the hypotenuse, here with length other two sides are legs, here with lengths a and a special case first here, an isosceles righttriangle. As needed, point out that good pieces mightbe formed if they draw lines through the smallersquares parallel to edges of the largest square.[Language]A dissectionis the result of separatingsomething into 1 Using the Dissection of Squares worksheets orthe Sketchpad demonstration will speed the investiga-tion, but the use of many different triangles drawn bythe students strengthens the inductive constructions are quicker with patty paper thanwith compass and straightedge.

5 It is also easy tocreate several examples using geometry 2As needed, remind students that the legs arethe sides other than the hypotenuse, so the longerleg is not the hypotenuse. Suggest that studentsminimize clutter by making these diagonals verylight or by drawing only the portion near the centerof the 4 Ask students to take care in drawing andcutting out pieces so they will fit together may want to tape the pieces OUTLINEOne day:15 minInvestigation5 minSharing10 minExamples15 minClosing and ExercisesMATERIALS construction tools scissors Pythagorean Theorem (W) for One step Dissection of Squares (W),optional Sketchpad demonstration ThreeTriangles,optionalTEACHINGMany students may already knowthe Pythagorean Theorem as a2 b2 c2. In this lesson theyreview what the letters stand forand discover proofs showing whythe relationship holds for allright students attention toImproving Your Visual ThinkingSkills on page 454.

6 Ask what theycan conclude about right trian-gles, and help them state thePythagorean Theorem usingareas of squares and the termshypotenuseand the InvestigationOne stepHand out a copy of thePythagorean Theorem worksheetto each group. Challenge studentsto cut up one or both of thesmaller squares and assemble thepieces on top of the largestsquare. As you circulate, youmight remind students of theproblem-solving technique ofHistoryPythagoras of Samos (ca. 569 475 ),depicted in this statue, is often described as the first pure mathematician. Samos was aprincipal commercial center of Greece and islocated on the island of Samos in the AegeanSea. The ancient town of Samos now lies inruins, as shown in the photo at , none of Pythagoras s writings stillexist, and we know very little about his life.

7 Hefounded a mathematical society in Croton, inwhat is now Italy, whose members discovered irrational numbers and the fiveregular solids. They proved what is now called the Pythagorean Theorem ,although it was discovered and used 1000 years earlier by the Chinese andBabylonians. Some math historians believe that the ancient Egyptians alsoused a special case of this property to construct right theoremis a conjecture that has been proved. Demonstrations like the one in the investigation are the first step toward proving the Pythagorean it or not, there are more than 200 proofs of the Pythagorean Scott Loomis s Pythagorean Proposition,first published in 1927, containsoriginal proofs by Pythagoras, Euclid, and even Leonardo da Vinci and U. James Garfield. One well-known proof of the Pythagorean Theorem is included below.

8 You will complete another proof as an Proof: The Pythagorean TheoremYou need to show that a2 b2equals c2for the right triangles in the figure at area of the entire square is a b 2or a2 2ab b2. The area of any triangleis 12 ab,so the sum of the areas of the four triangles is area of thequadrilateral in the center is a2 2ab b2 2ab,or a2 the quadrilateral in the center is a square then its area also equals nowneed to show that it is a square. You know that all the sides have length c,but you also need to show that the angles are right angles. The two acute angles in the right triangle, along with any angle of the quadrilateral, add up to 180 . The acute angles in a right triangle add up to 90 . Therefore the quadrilateral anglemeasures 90 and the quadrilateral is a square.

9 If it is a square with side length c,then its area is ,a2 b2 c2, which proves the Pythagorean Theorem . babababaccccThe Pythagorean TheoremIn a right triangle, the sum of the squares of the lengths of the legs equals thesquare of the length of the hypotenuse. Ifaand bare the lengths of the legs,and cis the length of the hypotenuse, then ?.a2 b2 c2C-82 Step 5 State the Pythagorean IDEASYou might make a transparencyof the Dissection of Squaresworksheets for students to use in presenting their about symmetry in thedissected square on the hypot-enuse. The method of the Inves-tigation gives 4-fold rotationalsymmetry.[Ask] What if the triangle isn t a right triangle? Do you thinkthere s still a relationship amongthe lengths of the sides? You neednot answer this question now;it s addressed later.

10 [Link]ThePythagorean Theorem is a specialcase of the Law of Cosines c2 a2 b2 2abcosC,where Cis the angle oppositeside c ;when m C 90 , wehave cos C 0.[Ask] What is the longest side ofa right triangle? Is it the same asthe longest leg? [The hypotenuse,not the longest leg, is the longestside.] If you ask why the longestside is always the hypotenuse andwhat can be said about the longerof the two legs, you can reviewthe Triangle Inequality Conjectureand the side -Angle InequalityConjecture.[Ask] What is a Theorem ? [It s aconjecture that has been proveddeductively within a deductivesystem.] So far in this course noaxiom system has been devel-oped, so there are no real theo-rems; this conjecture is called atheorem in this book because ithas been proved within an axiomsystem and it s so well known bythat students are not very familiarwith the Pythagorean Theorem ,ask how this Theorem about areasof squares might be used tocalculate lengths.