THE PYTHAGOREAN THEOREM - PDF Free Download

THE PYTHAGOREAN THEOREM The Pythgoren Theorem is one of the most well-known nd widely used theorems in mthemtis. We will first look t n informl investigtion of the Pythgoren Theorem, nd then pply this theorem to find missing sides of right tringles s well s the distne etween two points. Finlly, we will provide proofs of the Pythgoren Theorem tht pply to vriety of lerning styles. Let us now turn to the lssi illustrtion of the Pythgoren Theorem, nd perform some mesurements s n investigtion. (The digrms nd mesurements from this illustrtion hve een reted using The Geometer s Skethpd softwre.) First, we rete right tringle ABC. The onvention for nming sides is tht side is loted ross from A, side is loted ross from B, nd side is loted ross from C. We then find the length of its sides, rounded to the nerest hundredth. B =.41 m = 4.00 m = 4.66 m C A Next, we onstrut squres on the three sides of ABC. We then mesure the re of eh of the squres (to the nerest hundredth), s shown elow. Wht do you notie? B Are of Squre t side = 5.80 m Are of squre t side = 15.96 m Are of squre t side = 1.76 m C A

Notie tht the sum of the res of the two smller squres equls the re of the lrgest squre; this leds us to the more forml wording of the Pythgoren Theorem: The Pythgoren Theorem In right tringle, the sum of the squres of the lengths of the legs equls the squre of the length of the hypotenuse. In other words, if nd represent the lengths of the legs of right tringle, nd represents the length of the hypotenuse, the Pythgoren Theorem sttes tht: + = Exmples Find x. Write ll nswers in simplest rdil form. 1. Solution: The lengths of the legs re 6 nd 8, nd the x 8 length of the hypotenuse is x, so 6 + 8 = x 36 + 64 = x 100 = x 6 10 = x x = 10. Solution: 11 The lengths of the legs re 7 nd x, nd the x length of the hypotenuse is 11, so 7 + x = 11 49 + x = 11 7 x = 7 x = 7 x = 36 x = 6

3. Solution: 1 The lengths of the legs re x nd 3 5, nd 3 5 the length of the hypotenuse is 1, so x ( 3 5) 1 ( ) x + = x + = 3 5 3 5 144 x + 9 5 = 144 x + 45 = 144 x = 99 x = 99 x = 9 11 x = 3 11 Exerises Find x. Write ll nswers in simplest rdil form. 1. 13 x 5. x 8 4 3. 5 5 3 x

The Pythgoren Theorem nd the Distne Formul The Pythgoren Theorem n e used to find the distne etween two points, s shown elow. Exmples 1. Use the Pythgoren Theorem to find the distne etween the points A(, 3) nd B(7, 10). Write your nswer in simplest rdil form.. Use the Pythgoren Theorem to find the distne etween the points A(-3, 4) nd B(5, -6). Write your nswer in simplest rdil form. Solution to #1: We first plot the points A(, 3) nd B(7, 10) on the oordinte plne. We wnt to find the distne AB. Next, we drw right tringle ABC tht hs hypotenuse AB, s shown elow. 11 10 9 8 7 6 5 4 3 1 y A (, 3) B (7, 10) C (7, 3) -4-3 - -1-1 1 3 4 5 6 7 8 9 10 11 1 - -3 x We n esily find the lengths of the legs of the tringle: AC = 5 (We n quikly ount the units or tke the solute vlue of the differene of the x-oordintes: 7 = 5, or equivlently 7 = 5.)

BC = 7 (We n quikly ount the units or tke the solute vlue of the differene of the y-oordintes: 10 3 = 7, or equivlently 3 10 = 7.) We n now use the Pythgoren Theorem: 5 + 7 = ( AB) 5 + 49 = ( AB) 74 = ( AB) AB = 74 (Note: This squre root n not e simplified ny further. For dditionl informtion on simplifying squre roots, refer to the Irrtionl Numers tutoril in the ppendix ) Solution to #: We first plot the points A(-3, 4) nd B(5, -6) on the oordinte plne. We wnt to find the distne AB. Next, we drw right tringle ABC tht hs hypotenuse AB, s shown elow. A (-3, 4) 7 6 5 4 3 1-8 -7-6 -5-4 -3 - -1-1 1 3 4 5 6 7 8 - -3-4 -5-6 C (-3, -6) -7 B (5, -6) y x

We n esily find the lengths of the legs of the tringle: BC = 8 (We n quikly ount the units or find the solute vlue of the differene of the x-oordintes: 5 ( 3) = 8, or equivlently ( 3) 5 = 8.) AC = 10 (We n quikly ount the units or find the solute vlue of the differene of the y-oordintes; ( 6) 4 = 10, or equivlently 4 ( 6) = 10.) We n now use the Pythgoren Theorem: 8 + 10 = ( AB) 64 + 100 = ( AB) 164 = ( AB) AB = 164 AB = 4 41 AB = 41 (Note: For dditionl informtion on simplifying squre roots, refer to the Irrtionl Numers tutoril in the ppendix ) Now tht we hve used the Pythgoren Theorem, we will solve the sme prolems using the distne formul insted. The distne formul is shown elow. (An informl derivtion will e shown lter in this setion.) The Distne Formul Given two points Ax ( 1, y 1) nd B( x, y ), the distne d etween A nd B is ( ) ( ) 1 1 d = x x + y y Exmples 1. Use the distne formul to find the distne etween the points A(, 3) nd B(7, 10). Write your nswer in simplest rdil form.. Use the distne formul to find the distne etween the points A(-3, 4) nd B(5, -6). Write your nswer in simplest rdil form.

Solution to #1: ( ) ( ) 1 1 d = x x + y y ( 7 ) ( 10 3) d = + d = 5 + 7 d = 5 + 49 d = 74 Solution to #: ( 1) ( 1) ( 5 ( 3) ) ( 6 4) d = x x + y y d = + ( ) 8 10 d = + d = 64 + 100 d = 164 = 4 41 = 41 Notie how the numers in this solutions ove orrespond to those in the previous set of exmples where we used the Pythgoren Theorem diretly. We will now show n informl derivtion of the distne formul. x y x represents the length of one leg of the tringle. 1 y represents the length of the other leg of the tringle. 1 Sine d represents the hypotenuse of the tringle, we n use the Pythgoren Theorem nd otin the following formul: ( 1 ) ( 1 ) x x + y y = d ( ) ( ) 1 + 1 = x x y y d ( ) ( ) 1 1 d = x x + y y Exmples 1. Given the points A(3, 4) nd B(7, 1), ) Use the Pythgoren Theorem to find the distne from A to B. Write your nswer in simplest rdil form. ) Verify your nswer in prt () y using the distne formul.

. Given the points A(-1, 7) nd B(-4, -), ) Use the Pythgoren Theorem to find the distne from A to B. Write your nswer in simplest rdil form. ) Verify your nswer in prt () y using the distne formul.

Proofs of the Pythgoren Theorem We will now present four proofs of the Pythgoren Theorem. These prtiulr proofs hve een hosen euse s group, they ommodte vriety of lerning styles. The lerning styles re desried elow. Visul Visul lerners lern y seeing. They lern est from digrms, hrts, detiled written explntions, nd y tking detiled notes themselves. Auditory Auditory lerners lern y listening. When deling with written informtion, they lern muh etter when they n her the informtion eing explined to them. Ttile/Kinestheti Ttile/Kinestheti lerners lern y doing. They lern est when they n prtiipte in hnds-on tivities nd demonstrtions. After working through eh proof in detil, we will disuss the lerning style(s) tht pply most to tht prtiulr proof. Proof #1 The following proof is sed on the proof y Pythgors, ut we will first set up the proof in detil y sking the reder to help onstrut some digrms. We egin with the lssi illustrtion of the Pythgoren Theorem. We wish to show tht + =, or in other words, (the re of the squre t side ) + (the re of the squre t side ) = (the re of the squre t side ) B C A

We will now exmine the omined re of eh of the following groups of polygons. Group 1: The squre t side, the squre t side, nd four opies of the right tringle (with legs nd ). Group : The squre t side nd four opies of the right tringle (with legs nd ).

Proof #1: Ativity 1. Cut out the six piees from Group 1 elow.. Rerrnge the six piees nd ple them in suh wy tht they fit extly into the following squre of side length +.