11 Pythgors Theorem Pythgors Theorem. onverse of Pythgors theorem. Pythgoren triplets Proof of Pythgors theorem nd its converse Problems nd riders bsed on Pythgors theorem. This unit fcilittes you in, stting Pythgors theorem proving Pythgors theorem logiclly. stting converse of Pythgo rs theorem. pro ving converse of Pythgo rs theorem logiclly. explining the mening of Pythgoren triplets. resoning deductively nd prove the riders bsed on Pythgors theorem nd its converse. nlysing nd solving rel life problems bsed on pythgors theorem. Pythgors (569-79., Greece) Pythgors, pupil of Thles is perhps best known for the theorem on right ngled tringle which bers his nme. He helped to devlop the study of geometry in logicl wy by using undefined terms, definitions, postultes nd logicl deductions. The influence of his school in the fields of mthemtics, science, music, religio n nd philosophy is pprent even tody. There is geometry in the humming of the strings, there is music in the spcing of spheres. -Pythgors
70 UNIT-11 Let us recll some of the properties of right ngled tringle which we hve lredy lernt. 1. The side opposite to right ngle is clled hypotenuse.. In right ngled tringle, hypotenuse is the longest side. 3. In n isosceles right ngled tringle, ech cute ngle is 5.. re of right ngled tringle is hlf the product of the sides contining the right ngle. 5. The perpendiculr drwn from the right ngled vertex to the hypotenuse divides the tringle into two similr tringles which re similr to the given right ngled tringle. Now, let us lern one more very interesting property bout right ngled tringles. To understnd the property, do the following ctivity. Hypotenuse onstruct with = cm, = 3cm, = 5cm. Observe tht will be exctly 90. onstruct squres on ll the three sides nd divide them into squres of sides 1cm ech. Now, count the number of smll squres on ll the three sides. They re 9, 16 nd 5. If we dd 9 nd 16, we get 5. Repet this ctivity for right ngled tringles with sides, {6cm, 8cm nd 10cm}, {5cm, 1cm, 13cm} etc. Wht do we infer from this? We cn infer tht, "In right ngled tringle, the squre on the hypotenuse is equl to the sum of the squres on the other two sides". This sttement which gives the reltionship between res of sides of right ngled tringle is clled Pythgors theorem, nmed fter the Greek mthemticin Pythgors, who lived round 500. This theorem which is used in mny brnches of mthemtics hs ttrcted the ttention of mny mthemticins nd tody we hve hundreds of vrieties of proofs for it. The sttement of Pythgors theorem cn be proved prcticlly by nother wy. This ws given by Henry Perigl (1830). Hence, it is clled Perigl dissection. Study the ctivity given below nd do it in groups. onstruct, right ngled t on crd bord s shown in the figure. rw squres on, nd. In these squres, do the constructions s shown in the figure nd steps mentioned below. Mrk the middle point (P) of the squre drwn on the longest side contining the right ngle, i.e.. (This cn be obtined by joining the digonls of the squre) Through P drw E nd FG E. cm cm cm 3 cm cm 5 c 3 cm 3 cm P 5 cm m 3 cm 5 cm 5 cm
Pythgors Theorem 71 Mrk the qudrilterls formed s 1,,3 nd, nd the squre on s 5. (Observe the figure) ut the qudrilterls 1,,3,, nd 5. rrnge them on the squre drwn on the hypotenuse. (observe the figure) Wht conclusion we cn drw bout the res of the squres on sides of the right ngled tringle? Pythgors Theorem In right ngled tringle, the squre on the hypotenuse is equl to the sum of the squres on the other two sides. t : In, = 90 To Prove : + = onstruction : rw. Proof : Sttement Reson ompre nd, 90 ( t nd construction) is common. ( Equingulr tringles) = =....(1) ompre nd, ( similrity criteri) 90 ( t nd construction) is common [ Equingulr Tringles] y dding (1) nd () we get + + =....() = (. ) + (. ) = ( + ) [ similrity criteri] + =. = [ + = ) + = QE
7 UNIT-11 Know this! udhyn, n ncient Indin mthemticin in his Shulv sutrs composed during 800 sttes tht, "The digonl of rectngle produces both res of which its length nd bredth produce seprtely". d = l + b For this reson, this theorem is lso referred to s the 'udhyn Theorem'. l b d b l lternte proof for Pythgors theorem using re of trpezium t : In, 90 To prove : + c = b onstruction : Extend to such tht, =. t, drw nd mrk point E on it such tht, Join E, nd E,. E =. Proof : Sttement Reson c b c E ompre nd E = ( onstruction) = E 90 ( t nd construction) = E ( onstruction) E ( SS) 1) = E [ PT] ) = E nd 3) = E In, + = 90 [ = E ] E + = 90 [ djcent ngles of liner pir] E = 90
Pythgors Theorem 73 re of E = re of + re of E + re of E ½( + c)( + c) = ½c + ½b + ½c ( + c) = c + b [by cncelling 1 throughout] + c + c = c + b + c = b QE Know this To-dy we hve vrieties of proofs for Pythgors Theorem. It is sid tht there re more thn 300 proofs, some of them re proved bsed on trigonometry, co-ordinte geometry, vectors etc. Try to collect some of them nd discuss in groups. ILLUSTRTIVE EXMPLES Numericl problems bsed on Pythgors theorem Exmple 1: In right ngled, = 90, = 17cm nd = 8cm, find. Sol. Given, in, 90 Exmple : In, = + Pythgors theorem] = - = 17-8 = 89-6 = 5 = 5 15cm = 5, M, M = cm nd = 7cm. Find the length of. Sol. In M, M 90, M 5 M 5 8cm 17cm? M is n isosceles right ngled tringle M = M = cm M = M = 7 = 3cm M = 3cm In M, = M + M [ Pythgors theorem] = + 3 = 16 + 9 = 5 = 5 5cm
7 UNIT-11 Exmple 3 : In the rectngle WXYZ, XY + YZ = 17cm nd XZ + YW = 6 cm, clculte the length nd bredth of the rectngle. Sol. XZ + YW = 6 cm d 1 + d = 6 cm d = 6cm [ d 1 = d ] d XZ = 13 cm = YW = 13cm Let length = XY = x cm bredth = XW = (17 x)cm W X Z Y In WXY, WX + XY = WY [ pythgors theorem] (17 x) + x = 13 (89 3x + x ) + x = 169 (x 3x + 10 = 0) x 17x + 60 = 0 x 1x 5x + 60 = 0 x(x 1) 5(x 1) = 0 (x 1)(x 5) = 0 x 1 = 0 or x 5 = 0 Length x = 1 or x = 5 = 1cm, bredth = 5cm Exmple : In the, the hypotenuse is greter thn one of the other side by units nd it is twice greter thn the nother side by 1 unit. Find the mesure of the sides. Sol. Let = x nd = y = x + or = y + 1 x + = y + 1 x + 1 = y x 1 = y + = Pythgors theorem) x + y = (y + 1) x 1 x = y + y + 1 x x 1 x = x 1 x 1 1 x ( + ) x or ( + 1) y y
Pythgors Theorem 75 1 x x x = ( 1) ( 1) x x x x + x + x + 1 = (x + x + 1) + (x + ) + x + x + x + 1 = x + 8x + + 8x + 8 + x + x 16x + 1 16 = 0 Exmple 5 : Sol. x 1x 15 = 0 (x 15)(x + 1) = 0 x =15 or x = 1 If x = 15, x + = 17 nd y = x 1 16 8 = 15 units, = 8 units, = 17 units. n insect 8 m wy from the foot of lmp post which is 6m tll, crwls towrds it. fter moving through distnce, its distnce from the top of the lmp post is equl to the distnce it hs moved. How fr is the insect wy from the foot of the lmp post? [hskrchry's Leelvthi] istnce between the insect nd the foot of the lmp post = = 8m. The height of the lmp post = = 6m. fter moving distnce, let the insect be t, Let = = x m. = (8 - x) m. In, 90 1 6 =x 8-x x = + ( Pythgors theorem) x = 6 + ( 8- x) x = 36 + 6-16x + x 16x = 100 x = 100 6.5 16 = 8 x = (8 6.5) = 1.75 m The insect is 1.75 m. wy from the foot of the lmp post. Note : We cn lso consider = x, then = = (8 x) =
76 UNIT-11 Riders bsed on Pythgors Theorem. Exmple 6 : In the given figure,, Prove tht + = + Sol. In, 90 = + ( Pythgors theorem)...(1) In 90 = + ( Pythgors theorem)...() Subtrcting (1) from (), we get Exmple 7 : In, (i) = + - - = - + = + = 60 (ii). erive n expression for in terms of nd. Sol. onst: rw E such tht E 60 E is n equilterl [ t nd construction] E = = E E is n isosceles E 10 nd E 30 E E = E = E = E [ xiom - 1] = Now in, = + [ pythgors theorem] = ( ) + ( ) [ pythgors theorem] = +. + [ = ] = +. =. + = ( ) + = ( ) + [ = ) =. + or = +. y this we hve expressed interms of nd only. Exmple 8 : erive the formul for height nd re of n equilterl tringle. Sol. In the equilterl, let = = = '' units. rw Let = 'h' units
Pythgors Theorem 77 = = = [ RHS theorem] Now there re two right ngled tringles, nd. In, = 90 [ ] = + [ Pythgors theorem] = h + = h + = h 1 = h 3 = h Tke squre root on either sides. 3 3. = h = h h= 3 3 Height of n equilterl tringle of side units, h = Let us now find the re of equilterl. re of = 1 bse height = 1 3 = 3 re of n equilterl tringle, = 3
78 UNIT-11 Exmple 9 : Equilterl tringles re drwn on the sides of right ngled tringle. Show tht the re of the tringle on the hypotenuse is equl to sum of the re of tringles on the other two sides. Sol. t : In, 90 To prove : Equilterl les P, Q nd R re drwn on sides, nd respectively. re ( P) + re ( Q) = re ( R) Proof : P Q R [ Equingulr les re similr] P 3x x 3 + y 3 y Q x y x y R onsider re ( P) re ( R) re( Q) re ( R) re( P) re( Q) re( R) = re ( P) re( Q) re( R) = [ Theorem] [ pythgors theorem = + ] re ( P) re( Q) re( R) = 1 re ( P) + re ( Q) = re ( R) lternte Proof : P + Q = 3 3 3 x y x y Exmple 10 : 3 3 R = x y x y P + Q = R In, is point on such tht : = 1 : nd is n equilterl tringle Prove tht = 7 Sol. t: In, : = 1 : In, = = To Prove: = 7 onstruction: Proof: rw E In, E = E = nd E = 3 3 re of equilterl le= side 3 E
Pythgors Theorem 79 In E, E 90 [ onstruction] = E + E [ Pythgors theorem] = 3 3 5 = = = 3 5 3 5 8 = = 7 = 7 EXERISE 11.1 Numericl problems bsed on Pythgors theorem. 1. The sides of right ngled tringle contining the right ngle re 5cm nd 1cm, find its hypotenuse.. Find the length of the digonl of squre of side 1cm. 3. The length of the digonl of rectngulr plyground is 15m nd the length of one side is 75m. Find the length of the other side. L. In LW, LW 90, LN 90 nd LW = 6cm, LN = 6cm nd N = 8 cm. lculte the length of W. 6 cm N 8cm 6cm 5. door of width 6 meter hs n rch bove it hving height of meter. Find the rdius of the rch. W h = m 6m 6. The sides of right ngled tringle re in n rithmetic progression. Show tht the sides re in the rtio. 3 : : 5. 7. pecock on pillr of 9 feet height on seeing snke coming towrds its hole situted just below the pillr from distnce of 7 feet wy from the pillr will fly to ctch it. If both posess the sme speed, how fr from the pillr they re going to meet?
80 UNIT-11 Riders bsed on Pythgors theorem. M 8. In MGN, MP GN. If MG = units, MN = b units, GP = c units nd PN = d units. b Prove tht ( + b)( b) = (c + d)(c d). G c P d N 9. In L, L 90 nd LM L Prove tht L L M M 10. In, = 90 0,. If = 'c' units, = '' units, = 'p' units, = 'b' units. 1 1 1 Prove tht c p. c p b M Pythgoren triplets You hve lernt tht the mesures of three sides of right ngled tringle re relted in specil wy. If the three numbers, which re the mesures of three sides of right ngled tringle re nturl numbers then they re clled Pythgoren triplets. Some of them re listed in the tble given below. study them. 3,, 5 6, 8, 10 5, 1, 13 15, 36, 39 7,, 5 1, 8, 50 11, 60, 61, 0, ompre ech of the Pythgoren triplets in column with the corresponding triplet in column. We cn conclude tht; if ( x, y, z) is py thgore n triplet, then (kx, ky, kz) is lso Pythgoren triplet where k N. Know this! Pythgoren triplets cn be found using the following generl form. For nturl number: n, (n 1), (n + 1) Here 'n' my be even or odd. 1 For odd nturl numbers: n, ( 1), n 1 ( n 1), Here n is odd where n N. From the bove generl forms ny number of pythgoren triplets cn be generted by giving vlues to 'n'.
Pythgors Theorem 81 We hve lernt tht squre number cn be equl to sum of two squre numbers. oes this type of reltionship pply to cubes or other powers? Know this! Fermt's lst theorem [Pierre de Fermt ( 1601-1665), French mthemticin] "It is impossible to write - cube s the sum of two cubes. - fourth power s the sum of fourth powers" or In generl, "it is impossible to write ny power beyond the second s the sum of two similr powers." There re no, b, c, N for which n + b n = c n where n N nd n > ndrew Wiles (orn in cmbridge, Englnd - 1963) sw this problem, when he ws ten yers old. Thirty yers lter nd with seven yers of intense work ndrew wiles, proved Fermt's lst theorem. onverse of Pythgors Theorem Let us now lern the converse of pythgors theorem. For this, consider the two exmples. Exmple 1 cm 3 cm Exmple cm P.3 cm Let be tring le with = 5cm, = cm nd = 3cm. Imgine squres on ll the three sides nd find their res. Here, the longest side is nd its length is 5cm. The re of the squre on the longest side is 5 sq. cm The sum of the squres on the other two sides nd will be (16 + 9) sq cm which is lso 5 sq. cm. Now mesure which is opposite to the longest side. you will find tht = 90. 5 cm So, is right ngled tringle, right ngled t the vertex. Q 5 cm R Let PQR be nother tringle with QR = 5cm, PQ = cm nd PR =.3 cm. Imgine squres on ll the three sides nd find their res. Here, the longest side is QR nd its length is 5 cm. The re of the squre on the longest side QR is 5 sq. cm. The sum of the squres on the other two sides PQ nd PR will be (16 + 18.9) sq cm = 3.9 sq. cm which is not equl to the re of the squre on the longest side QR. Now mesure QPR which is opposite to the longest side QR, you will find tht QPR=7 but not 90. So, PQR is not right ngled tringle.
8 UNIT-11 So, wht is the bsic condition for tringle to be right ngled tringle? ompre the res on the sides of two tringles in the bove exmples nd try to figure out the necessry condtion. The condition is: "If the squre on the longest side of tringle is equl to the sum of the squres on the other two sides then those two sides contin right ngle." This is converse of Pythgors theorem. It ws first mentioned nd proved by Euclid. Now let us prove the converse of Pythgors theorem logiclly. onverse of pythgors theorem "If the squre on the longest side of tringle is equl to the sum of the squres on the other two sides, then those two sides contin right ngle." t : In, = + To prove : = 90 onstruction : rw perpendiculr on t. Select point on it such tht, =. Join '' nd ''. Sttement Reson Proof : In, = 90 ( onstruction) = + ( Pythgors theorem) ut in, = + ( t) = = ompre nd = = is common ( Proved) ( onstruction) ( SSS) = ( PT) = = 90 QE
Pythgors Theorem 83 omprision of Pythgors theorem nd its converse. Pythgors theorem "In right ngled tringle, the squre on the hypotenuse is equl to the sum of the squres on the other two sides. onverse of Pythgors theorem "If the squre on the longest side of tringle is equl to the sum of the squres on the other two sides, then those two sides contin right ngle." t: In, = 90 To prove: = + t: In, = + To prove: = 90 Now observe the following tble. Study the reltionship between the res of three sides of tringle. c b c b c b In, = 90 = + b = c + * In, < 90 < + b < c + * In, > 90 > + b > c + onverse If b = c + then = 90 If b < c + then < 90 If b > c + then > 90 iscuss The Pythgors theorem hs two fundmentl spects, where one is bout res nd the other is bout lengths. This lndmrk theorem connects two min brnches of mthemtics - Geometry nd lgebr.
8 UNIT-11 ILLUSTRTIVE EXMPLES Exmple1 : Verify whether the following mesures represent the sides of right ngled tringle. () 6, 8, 10 Sol. Sides re : 6, 8, 10 onsider the res of squre on the sides : 6, 8, 10 i.e., 36, 6, 100 onsider the sum of res of squres on the two smller sides : 36 + 6 = 100 6 + 8 = 10 We observe tht, squre on the longest side of the tringle is equl to the sum of squres on the other two sides. y converse of Pythgors theorem, those two smller sides must contin right ngle. onclusion: The sides 6, 8 nd 10 form the sides of right ngled tringle with hypotenuse 10 units nd 6 nd 8 units s the sides contining the right ngle. Note: Without ctully constructing the tringle for the given mesurements of sides it is now possible to sy whether the sides represent the sides of right ngled tringle using converse of Pythgors theorem. (b) 1,, 3 Sides re : 1, 3, onsider the res of squres on the sides : 1, 3, i.e., : 1, 3, onsider the sum of the res of squres on the two smller sides : 1 + 3 = 1 + 3 = We observe tht the squre on the longest side ( units) is equl to the sum of the squres on the other two sides. y converse of pythgors Theorem, these two smller sides must contin right ngle. onclusion: 1,, hypotenuse nd 1 nd (c), 5, 6 3 forms the sides of right ngled tringle with units s 3 units s sides contining the right ngle. Sides re :, 5, 6 res of squres on the sides :, 5, 6 i.e., : 16, 5, 36
Pythgors Theorem 85 Sum of res of squres on the two smller sides : 16 + 5 = 1 + 5 6 We observe tht the squre on the longest side of the tringle is not equl to the sum of the squres on the other two sides. y converse of Pythgors theorem, these two sides cnnot contin right ngle. Hence,, 5, nd 6 cnnot form the sides of right ngled tringle. Exmple : In the qudrilterl, 90 nd = ( + + ). Prove tht : 90 Proof: In, 90 [ t] = + [ Pythgors theorem] ut = ( + )+ [ t] = + [ by dt + = ] = 90 [ converse of Pythgors theorem] EXERISE 11. 1. Verify whether the following mesures represent the sides of right ngled tringle. (), 3, 5 (b) 6 3, 1, 6 (c) n, n 1, n 1 (d) x 1, x, x + 1 (e) x 1 x 1,, x (f) m n, mn, m + n. In, + b = 18 units, b + c = 5 units nd c + = 17 units. Wht type of tringle is? Give reson. c b
86 UNIT-11 3. In,, = nd = 3, Prove tht 90.. The shortest distnce P from point '' to QR is 1 cm. Q nd R re respectively 15 cm nd 0cm from '' nd on opposite sides of P. Prove tht QR 90 15 cm 1 cm 0 cm 5. In the isosceles, =, = 18 cm,, = 1 cm, is produced to 'E' nd E = 0cm. P 1 cm 0 cm R Prove tht E 90 18 cm E 6. In the qudrilterl, 90, = 9cm, = = 6cm nd = 3cm, Prove tht 0 90 3 cm 6 cm 6 cm 9 cm P 7. is rectngle. 'P' is ny point outside it such tht P + P = +. Prove tht P 90 Pythgors theorem Sttement of Pythgors theorem onverse of Pythgors theorem Pythgoren triplets Logicl proof Riders EXERISE 11.1 NSWERS 1] 13 cm ]1 cm 3] 100m ] cm 5] 3.5m 6] 3: : 5 7] 1 ft