Apply the Pythagorean Theorem


 Nathan Doyle
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1 8. Apply the Pythgoren Theorem The Pythgoren theorem is nmed fter the Greek philosopher nd mthemtiin Pythgors ( B.C.E.). Although nient texts indite tht different iviliztions understood this property of right tringles, Pythgors proved tht it pplies to ll right tringles. hypotenuse the longest side of right tringle the side opposite the 90 ngle If right tringle is lelled s shown, then the re of the lrge squre drwn on the hypotenuse is, while the res of the other two squres re nd. Aording to the Pythgoren reltionship, the re of the squre drwn on the hypotenuse is equl to the sum of the res of the squres drwn on the other two sides. Pythgoren theorem in right tringle, the squre of the length of the hypotenuse is equl to the sum of the squres of the lengths of the two shorter sides Tools grid pper ruler Therefore, the lgeri model for the Pythgoren reltionship is. This is known s the Pythgoren theorem. Investigte How n you illustrte the Pythgoren theorem? Method : Use Penil nd Pper. Construt ny right tringle. Lel the sides of your tringle using three different letters.. Mesure the length of eh side of your tringle. Indite these mesures on your digrm. 3. ) Clulte the re of the squre on the hypotenuse. ) Clulte the sum of the res of the squres on the two shorter sides. ) Write the Pythgoren theorem using your side lels. 48 MHR Chpter 8
2 4. ) Clulte the squre root of your nswer to step 3). ) Compre this vlue to the length of the hypotenuse. 5. Construt ny nonright tringle. Does the Pythgoren reltionship still hold? Does the reltionship from step 4, prt ), still hold? 6. Reflet Explin how this tivity illustrtes the Pythgoren theorem. Method : Use The Geometer s Skethpd. From the Edit menu, hoose Preferenes. Clik on the Units t. Set the preision to tenths for ll three oxes. Clik on the Text t nd hek For All New Points. Clik on OK.. Use the Strightedge Tool to rete ny ABC. 3. ) To mesure ABC, selet verties A, B, nd C, in tht order. From the Mesure menu, hoose Angle. ) To mesure the length of AB, selet line segment AB. From the Mesure menu, hoose Length. Repet for line segments BC nd CA. 4. ) Drg vertex of the tringle until ABC mesures 90. ) Selet the mesure mca. From the Mesure menu, hoose Clulte. Enter mca^, y seleting mca from the Vlues dropdown menu on the lultor. ) Selet mab nd mbc. From the Mesure menu, hoose Clulte. Enter mab^ mbc^. Tools omputers The Geometer s Skethpd softwre Go to links/priniples9 nd follow the links to n intertive proof of the Pythgoren theorem. Did You Know? To rete right ngle for mesuring lnd or uilding pyrmids, the nient Egyptins tied eqully sped knots in rope. They then tied the rope into loop nd strethed it to form tringle with knot t eh vertex. The only wy this works is in the rtio 3:4:5, resulting in right tringle. 8. Apply the Pythgoren Theorem MHR 49
3 d 5. ) Selet (mab ) nd (mbc ). From the Mesure menu, hoose Clulte. Evlute (mab) (mbc) y hoosing sqrt from the Funtions pulldown menu on the lultor. ) Compre this vlue to the length of side CA. 6. Drg vertex of the tringle so tht the mesure of ABC is no longer 90. Does the Pythgoren reltionship still hold? Does the reltionship from step 5) still hold? 7. Reflet Explin how this tivity illustrtes the Pythgoren theorem. Exmple Find the Hypotenuse The dvertised size of omputer or television sreen is tully the length of the digonl of the sreen. A omputer sreen mesures 30 m y.5 m. Determine the length of its digonl. Solution In the digrm, the digonl, d, is the hypotenuse. Apply the Pythgoren theorem. d d m d d 30 m Only the positive squre root needs to e used euse d 37.5 d is length. The length of the digonl of the omputer sreen is 37.5 m. Exmple Find One of the Shorter Sides Jenn is hnging light ul. She rests 4m ldder ginst vertil wll so tht its se is.4 m from the wll. How high up the wll does the top of the ldder reh? Round your nswer to the nerest tenth of metre. 40 MHR Chpter 8
4 Solution In this se, the ldder is the hypotenuse, with length of 4 m. The unknown side length is h. Apply the Pythgoren theorem. 4.4 h 6.96 h h Sutrt.96 from oth sides h 4.04 h Tke the squre root of oth sides. 3.7 h The ldder rehes 3.7 m up the wll, to the nerest tenth of metre. 4 m.4 m h Exmple 3 Clulte the Are of Right Tringle Clulte the re of the tringulr sil on the toy silot. 8 m m Solution The formul for the re of tringle is A h. The se,, nd the height, h, must e perpendiulr to eh other. For right tringle, the se nd the height re the lengths of the two shorter sides. First, use the Pythgoren theorem to find the length of the unknown side, Sutrt 64 from oth sides Tke the squre root of oth sides. 7.5 The length of side is pproximtely 7.5 m. Now, pply the formul for the re of tringle. A h (8)(7.5) 30 I n write the re formul for tringle in different wys: A h, A h, nd A 0.5h. h Liter onnetions The perpendiulr sides of right tringle re lled the legs of the tringle. The re of the sil is pproximtely 30 m. 8. Apply the Pythgoren Theorem MHR 4
5 Key Conepts The longest side of right tringle is the hypotenuse. The Pythgoren theorem sttes tht the squre of the length of the hypotenuse is equl to the sum of the squres of the lengths of the two shorter sides. An lgeri model representing the Pythgoren theorem is, where represents the length of the hypotenuse nd nd represent the lengths of the two shorter sides. You n use the Pythgoren theorem to lulte the length of n unknown side of right tringle. You n lulte the re of right tringle y using the formul A h, with the lengths of the two shorter sides s the se,, nd the height, h. If one of these dimensions is unknown nd you know the hypotenuse, pply the Pythgoren theorem to lulte the length of the unknown side. Then, use the re formul. h Communite Your Understnding C Desrie how you n use the Pythgoren theorem to determine the length of the digonl of the squre. 5 m d 5 m C Desrie how you n use the Pythgoren theorem to determine the distne etween two points on grid. y 4 A B x C3 Desrie how you would find the re of right tringle if you knew the lengths of the hypotenuse nd one of the other two sides. 5 m 3 m 4 MHR Chpter 8
6 Prtise For help with question, see Exmple.. Clulte the length of the hypotenuse in eh tringle. Round your nswers to the nerest tenth of unit, when neessry. ) ) 6 m m 8 m ) d) 5 m 4. m 7 m 5. m For help with question, see Exmple. 5 m. Clulte the length of the unknown side in eh tringle. Round your nswers to the nerest tenth of unit, when neessry. ) ) 4 m 0 m 7 m 8 m ) d) 9.5 m 5.5 m 8. m For help with question 3, see Exmple m 3. Determine the re of eh right tringle. Round your nswers to the nerest tenth of squre unit, when neessry. ) ) 7 m m 8 m 0 m 8. Apply the Pythgoren Theorem MHR 43
7 Connet nd Apply 4. Clulte the length of eh line segment. Round nswers to the nerest tenth of unit, when neessry. ) AB ) CD ) EF y A D E B C F x 5. Wht is the length of the digonl of omputer sreen tht mesures 8 m y m? 8 m m Did You Know? Bsell ws formlly introdued s medl sport t the 99 Summer Olympis in Brelon, Spin. Cnd mde its first pperne in this event in the 004 Summer Olympis. 6. A sell dimond is squre with sides tht mesure out 7 m. How fr does the seondse plyer hve to throw the ll to get runner out t home plte? Round your nswer to the nerest metre. 7 m 7. A squre ourtyrd hs digonl pths tht re eh 4 m long. Wht is the perimeter of the ourtyrd, to the nerest metre? 8. Brook is flying kite while stnding 50 m from the se of tree t the prk. Her kite is diretly ove the 0m tree nd the 5m string is fully extended. Approximtely how fr ove the tree is her kite flying? 9. Chpter Prolem Emily hs designed tringulr flower ed for the orner of her lient s retngulr lot. The ed is fened on two sides nd Emily will use order stones for the third side. The ed mesures m nd.5 m long the fened sides. How mny order stones, 30 m in length, will Emily need to edge the flower ed?.5 m m 44 MHR Chpter 8
8 Extend 0. A rdord ox mesures 40 m y 40 m y 30 m. Clulte the length of the spe digonl, to the nerest entimetre. 30 m spe digonl 40 m 40 m. The Spider nd the Fly Prolem is lssi puzzle tht originlly ppered in n English newspper in 903. It ws posed y H.E. Dudeney. In retngulr room with dimensions 30 ft y ft y ft, spider is loted in the middle of one ft y ft wll, ft wy from the eiling. A fly is in the middle of the opposite wll ft wy from the floor. If the fly does not move, wht is the shortest distne tht the spider n rwl long the wlls, eiling, nd floor to pture the fly? Hint: Using net of the room will help you get the nswer, whih is less thn 4 ft! spider fly ft 30 ft ft. A spirl is formed with right tringles, s shown in the digrm. ) Clulte the length of the hypotenuse of eh tringle, leving your nswers in squre root form. Desrie the pttern tht results. ) Clulte the re of the spirl shown. ) Desrie how the expression for the re would hnge if the pttern ontinued. 3. Mth Contest ) The set of whole numers (5,, 3) is lled Pythgoren triple. Explin why this nme is pproprite. ) The smllest Pythgoren triple is (3, 4, 5). Investigte whether multiples of Pythgoren triple mke Pythgoren triples. ) Sustitute vlues for m nd n to investigte whether triples of the form (m n, mn, m n ) re Pythgoren triples. d) Wht re the restritions on the vlues of m nd n in prt )? d 8. Apply the Pythgoren Theorem MHR 45
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