HPTER Mesurement nd Geometry Pythgors theorem nd surds In IE-EM Mthemtis Yer 8, you lernt out the remrkle reltionship etween the lengths of the sides of right-ngled tringle. This result is known s Pythgors theorem. The theorem sttes tht the squre of the length of the hypotenuse equls the sum of the squres of the lengths of the other two sides. In symols: + The onverse of this theorem is lso true. This mens tht if we hve tringle in whih the squre of one side equls the sum of the squres of the other two sides, then the tringle is right-ngled, with the longest side eing the hypotenuse. Pythgors theorem leds to the disovery of ertin irrtionl numers, suh s nd. These numers re emples of surds. In this hpter, we investigte the rithmeti of surds. Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
Review of Pythgors' theorem nd pplitions In Yer 8, we used Pythgors theorem to solve prolems relted to right-ngled tringles. Pythgors theorem nd its onverse In ny right-ngled tringle, the squre of the length of the hypotenuse equls the sum of the squres of the lengths of the other two sides. Emple Find the vlue of the unknown side in eh tringle. 9 + tringle with side lengths of,, whih stisfy + is right-ngled tringle. The right ngle is opposite the side of length. 9 + 40 68 68 4 40 lultors nd rounding 4 0 + 0 4 4 0 56 56 6 In Emple ove, the vlues for re perfet squres nd so we ould tke the squre root esily. This is not lwys the se. In Yer 8, we found the pproimte squre roots of numers tht re not perfet squres y looking up tle of squre roots. Insted of doing this, from now on we re going to use lultor. HPTER Pythgors theorem nd surds Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
Review of Pythgors' theorem nd pplitions When using lultor to find squre root, try to hve in mind rough ide of wht the nswer should e. For emple, 0 should e lose to, sine nd 44. Note: 0.40 orret to deiml ples. lultor gives the pproimte vlue of squre root to lrge numer of deiml ples, fr more thn we need. We often round off deiml to required numer of deiml ples. The method for rounding to deiml ples is s follows. Look t the digit in the third deiml ple. If the digit is less thn 5, tke the two digits to the right of the deiml point. For emple,.764 eomes.76, orret to deiml ples. If the digit is more thn 4, tke the two digits to the right of the deiml point nd inrese the seond of these y one. For emple,.455 eomes.46, orret to deiml ples. Emple lultor gives.7050808. Stte the vlue of orret to deiml ples. lultor gives 5.6067977. Stte the vlue of 5 orret to deiml ples. Stte the vlue of.697, orret to deiml ples..7, sine the digit,, in the third deiml ple is less thn 5. 5.4, sine the digit, 6, in the third deiml ple is more thn 4..697.70, sine the digit, 7, in the third deiml ple is more thn 4 nd 69 rounds to 70. The symol mens is pproimtely equl to. When pproimting, we should stte the numer of deiml ples to whih we hve rounded our nswer. Emple Find the length of the unknown side, orret to deiml ples. + 0 044 044. (orret to deimlples) 0 4 IE-EM MTHEMTIS YER 9 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
Review of Pythgors' theorem nd pplitions In the emple on the previous pge, 044 is the et length while. is n pproimtion to the length of the side. Emple 4 Determine whether or not the three side lengths given form the sides of right-ngled tringle. 4,, 40 4,8, The squre of the length of the longest side 40 600 The sum of the squres of the lengths of the other sides 4 + 576 + 04 600 Sine 40 4 +, the tringle is right-ngled. The squre of the length of the longest side 59 The sum of the squres of the lengths of the other sides 4 + 8 96 + 4 50 Sine 4 + 8, the tringle is not right-ngled. Emple 5 door frme hs height.7 m nd width m. Will squre piee of ord m y m fit through the doorwy? Let d m e the length of the digonl of the doorwy. Using Pythgors theorem, d +.7.89 d.89 d m.7 m.97 (orret to deiml ples) Hene the ord will not fit through the doorwy. m HPTER Pythgors theorem nd surds 5 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
R e v i e w o f P y t h g o rs ' t h e o rem n d pp l i t i o n s Eerise Emple Use Pythgors theorem to find the vlue of the pronumerl. 5 m m 8 m m m 6 m d m 4 m ES.6 m Use Pythgors theorem to find the vlue of the pronumerl. 5 m m d e 5 m m 45 m f 48 m m N L y m 50 m 70 m Use lultor to find, orret to deiml ples, pproimtions to these numers. Emple 74 m 6 m y m m 7 m m 8 m 5 m P Emple G. m Emple 40 m m 9 7 d 6 7 4 Use Pythgors theorem to find the vlue of the pronumerl. lulte your nswer first s squre root nd then orret to deiml ples. m FI 4 m 7 m m 5 m d 8 m e 7 m m m y m f 8 m 0 m m y m m m 4 m 7 m 6 I E - E M M T H E M T I S YER 9 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
Review of Pythgors' theorem nd pplitions g h 7 m m 6 m 8 m Emple 4 Emple 5 6 m 5 Determine whether or not the tringle with the three side lengths given is right-ngled. y m 6, 0, 4 0, 4, 6 4, 6, 7 d 4.5, 7, 7.5 e 6, 0, f 0,, 9 6 In the tle elow, the side lengths of right-ngled tringles re listed. opy nd omplete the tle, giving nswers s whole numer or squre root. Lengths of the two shortest sides of right-ngled tringle Length of the hypotenuse m 4m... 5m 6m... 4m... 9m d 6m 0 m... e... 7m 0 m 7 door frme hs height.8 m nd width m. Will squre piee of ord. m wide fit through the opening? 8 trdesmn is mking the wooden retngulr frme for gte. In order to mke the frme stronger nd to keep it squre, the trdesmn will put digonl piee into the frme s shown in the digrm. If the frme is. m wide nd m high, find the length of the digonl piee of wood, in metres, orret to deiml ples. 9 signwriter lens his ldder ginst wll so tht he n pint sign. The wll is vertil nd the ground in front of the wll is horizontl. The signwriter s ldder is 4m long. If the signwriter wnts the top of the ldder to e.8 m ove the ground when lening ginst the wll, how fr, orret to deiml ple, should the foot of the ldder e pled from the wll? 0 ot uilder needs to lulte the lengths of the stys needed to support mst on yht. Two of the stys ( nd D) will e the sme length, s they go from point on the mst to eh side of the ot, s shown in the digrm. The third sty (E) will e different in length s it goes from the point on the mst to the front of the ot.. m m Front elevtion Side elevtion D E HPTER Pythgors theorem nd surds 7 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
Review of Pythgors' theorem nd pplitions If 5m, D. m nd E.9 m, find the length, to the nerest entimetre, of: one of the side stys, or D the front sty, E stinless steel wire needed to mke the three stys s prt of design, n rtist drws irle pssing through the four orners (verties) of squre. If the squre hs side lengths of 4 m, wht is the rdius, to the nerest millimetre, of the irle? If the irle hs rdius of m, wht re the side lengths, to the nerest millimetre, of the squre? prent is sked to mke some srves for the lol Sout troop. Two srves n e mde from one squre piee of mteril y utting on the digonl. If this digonl side length is to e 00 m long, wht must e the side length of the squre piee of mteril to the nerest mm? girl plnned to swim stright ross river of width 5 m. fter she hd swum ross the river, the girl found she hd een swept 4 m downstrem. How fr did she tully swim? lulte your nswer, in metres, orret to deiml ple. 4 yhtsmn wishes to uild shed with retngulr se to store his siling equipment. If the shed is to e.6 m wide nd must e le to house 4.6 m mst, whih is to e stored digonlly ross the eiling, how long must the shed e? lulte your nswer, in metres, orret to deiml ple. 5 In tringle the line D is drwn perpendiulr to. h is the length of D nd is the length of D. Show tht the length of is 6. Find the re of tringle in two wys to show tht h 0. Use Pythgors theorem to find. 4 h 0 D 8 IE-EM MTHEMTIS YER 9 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
6 In digrms nd, we hve two squres with the sme side length, +. The sides re divided up into lengths nd s shown. Prove tht the shded prt of digrm is squre. The two shded prts of digrm re lso squres. Why? Digrm Digrm y looking refully t the two digrms, show tht the re of the shded squre in digrm is the sum of the res of the shded squres in digrm. Use the result of prt to prove Pythgors theorem. Simplifying surds This setion dels only with surds tht re squre roots. If is positive integer whih is not perfet squre then is lled surd. We will look t surds more generlly in Setion I. We will now review the si rules for squre roots. If nd re positive numers then: For emple: ( ) ( ) 7 7 5 5 5 5 7 The first two of these rules remind us tht, for positive numers, squring nd tking squre root re inverse proesses. When we write, we men. s in lger, we usully do not epliitly write the multiplition sign. HPTER Pythgors theorem nd surds 9 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
Simplifying surds Emple 6 Evlute: ( 6) ( 6) ( 6) 6 6 Emple 7 6 Evlute: 5 4 6 Emple 8 4 6 7 ( 6) 6 6 6 6 4 6 4 4 6 5 5 5 d 5 0 Evlute: 7 5 9 4 7 5 5 9 4 6 5 0 7 d 5 0 onsider the surd. We n ftor out the perfet squre 4 from, nd write: 4 4 (using ) 40 IE-EM MTHEMTIS YER 9 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
Simplifying surds Hene nd re equl. We will regrd s simpler form thn sine the numer under the squre root sign is smller. To simplify surd (or multiple of surd) we write it so tht the numer under the squre root sign hs no ftors tht re perfet squres other thn. For emple, In mthemtis, we re often instruted to leve our nswers in surd form. This simply mens tht we should not pproimte the nswer using lultor, ut leve the nswer, in simplest form, using squre roots, ue roots, et. This is lled giving the et vlue of the nswer. We n simplify surds diretly or in stges. Emple 9 Simplify the following. 8 08 8 9 9 In some prolems, we need to reverse this proess. Emple 0 Epress s the squre root of whole numer. 7 5 7 9 7 6 Emple 08 9 4 9 4 6 5 5 75 Use Pythgors theorem to find the vlue of. Give your nswer s surd whih hs een simplified. 5 4 6 HPTER Pythgors theorem nd surds 4 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
Simplifying surds Emple 6 Emple 6 Emple 7 + 9 + 9 8 8 9 The rithmeti of surds If nd re positive numers, then: ( ) ( 5) + ( ) 5 + 8 8 4 + 6 4 + 6 96 60 60 60 0 4 0 surd is in its simplest form if the numer under the squre root sign hs no ftors tht re perfet squres, prt from. To simplify surd, we write it so tht the numer under the squre root sign hs no ftors tht re perfet squres, prt from. For emple, 50 5. Eerise Evlute: ( 7) ( ) ( ) d ( ) Evlute: ( ) (4 ) (5 ) d ( 5) e ( 7) f (7 ) g ( ) h ( ) Epress s squre root of whole numer. 5 6 5 6 d e 7 f 5 g 4 h 0 9 4 IE-EM MTHEMTIS YER 9 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
Simplifying surds Emple 7,, d Emple 8 Emple 9 Emple 9 Emple 0 Emple 4 Epress s squre root of numer. 6 5 7 e 77 5 Evlute the produt. 4 f 40 8 g 6 4 h 6 5 7 d 6 5 7 e 5 f 7 5 6 g 4 h 7 4 6 Evlute: ( ) ( 5) ( 5) ( ) 9 d 46 d ( 7) ( ) e ( ) + ( ) f ( 5) + ( ) g 8 h i 7 Simplify eh of these surds. 8 45 d 4 e 7 f 44 g 50 h 54 i 0 j 98 k 6 l 60 m 6 n 68 o 75 p 99 q 8 r 4 8 Simplify eh of these surds. 7 80 d 88 e 48 f 80 g h 6 i 96 j 5 k 60 l 8 m 0 n 76 o 9 p 00 q 6 r 4 9 Epress eh of these surds s the squre root of whole numer. 6 7 d 6 e 4 5 f 5 7 g 4 h i 6 j 0 k 0 7 l 4 0 Evlute: e 4 5 5 f 5 6 5 7 g 5 d 6 Use Pythgors theorem to find the vlue of. Give your nswer s surd whih hs een simplified. 6 h 4 44 69 HPTER Pythgors theorem nd surds 4 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
S i mp l i f y i n g s u r d s d e 9 f 5 9 0 h i 5 50 G Find the length of the digonl D of the retngle D. Epress your nswer in simplest form. 0 ES g P D 6 squre hs side length. Find: the re of the squre the length of the digonl N L 4 D is squre. Find the vlue of. D FI 5 is n equilterl tringle with side length. X is the midpoint of. Find the length X. I E - E M M T H E M T I S 6 44 X YER 9 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
ddition nd sutrtion of surds We n sometimes simplify sums of surds. The sum 4 7 + 5 7 n e thought of s 4 lots of 7 plus 5 lots of 7 equls 9 lots of 7. This is very similr to lger, where we write 4 + 5 9. We sy 4 7 nd 5 7 re like surds sine they re oth multiples of 7. On the other hnd, in lger we nnot simplify 4 + 7y, euse 4 nd 7 y re not like terms. Similrly, it is not possile to write the sum of 4 nd 7 in simpler wy. They re unlike surds, sine one is multiple of while the other is multiple of. We n only simplify the sum or differene of like surds. Emple Simplify: 4 7 + 5 7 6 7 7 + 5 d 5 + 7 5 + 4 7 4 7 + 5 7 9 7 6 7 7 4 7 + 5 4 d 5 + 7 5 + 4 7 5 + 6 7 This nnot e simplified further sine 5 nd 7 re unlike surds. When deling with epressions involving surds, we should simplify the surds first nd then look for like surds. Emple Simplify: 8 + 4 8 7 + 5 + 0 8 + 4 8 + 4 7 + 5 + 0 + 5 + 5 + 4 5 This epression nnot e simplified further. HPTER Pythgors theorem nd surds 45 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
ddition nd sutrtion of surds Emple 4 Simplify: + 5 8 5 Emple 8 Simplify: 5 + + 5 0 0 5 6 0 5 0 5 5 7 5 ddition nd sutrtion of surds (Use ommon denomintor.) (Simplify the surds.) Simplify eh surd first, then look for like surds. (Use ommon denomintor.) We n simplify sums nd differenes of like surds. We nnot simplify sums nd differenes of unlike surds. Eerise 5 5 6 + 4 + 4 d 8 e 0 7 5 7 f 9 5 + 4 5 g + 4 h 5 i 45 + 50 Simplify: + + 4 6 6 8 6 0 5 5 6 5 d 5 + 6 e 7 8 7 + 5 7 f 0 + 9 0 7 0 g 6 + 4 6 + h 4 5 + 7 5 4 46 IE-EM MTHEMTIS YER 9 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
ddition nd sutrtion of surds Emple Emple 4 Simplify: + 4 + + 4 6 + + 9 7 6 d 5 4 + 4 6 + 4 + 6 e 6 7 4 + 4 4 7 7 f 5 4 5 + 7 4 Simplify: 8 + 8 50 + d 8 + e 7 + f 48 + g 5 + 45 h 8 + 6 i 8 + 4 + 8 5 Simplify: 7 50 48 + 8 + + 8 d + 4 75 e 00 + 50 f 4 5 4 0 45 g 54 + 4 h 7 48 + 75 i 45 + 80 5 j + + 7 k 000 40 90 l 5 + 8 + 6 Simplify: e 7 6 + 5 7 Find the vlue of if: f 4 + 6 + g 5 7 7 d + 5 7 7 5 h 7 6 8 80 45 4 54 8 Find the vlue of nd the perimeter. M 5 d 6 5 Z N O X Y HPTER Pythgors theorem nd surds 47 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
9 In the digrm to the right, 5, X, 5 5, YX 5. X Y Z d Y Y X Z 0 In the digrm to the right, find: X X Find nd the perimeter of the retngle. qudrilterl D hs side lengths 9, 6 nd D 8. D 90 o nd the digonl 67. Find the perimeter of the qudrilterl. D 9 8 6 67 Multiplition nd division of surds When we ome to multiply two surds, we simply multiply the numers outside the squre root sign together, nd similrly, multiply the numers under the squre root signs. similr rule holds for division. X 5 4 7 7 5 D D 48 IE-EM MTHEMTIS YER 9 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
D Multiplition nd division of surds Emple 5 Find 4 7. 4 7 8 4 (4 8, 7 4) We n stte the proedure we just used s generl rule: d d, where nd d re positive numers. Emple 6 Find: 5 5 5 7 49 4 5 5 5 7 5 (5 5, 5 7 5) 7 7 7 49 4 (49 4, 7) We n stte the proedure we just used s generl rule: d, d where nd d re positive numers nd 0. s usul, we should lwys give the nswer in simplest form. Emple 7 Find 5 6 0. 5 6 0 5 60 5 4 5 0 5 HPTER Pythgors theorem nd surds 49 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
D Multiplition nd division of surds The distriutive lw We n pply the distriutive lw to epressions involving surds, just s we do in lger. Emple 8 Epnd nd simplify (4 + ). (4 + ) 4 + 8 + 6 9 8 + 8 In lger, you lernt how to epnd rkets suh s ( + )( + d). These re known s inomil produts. You multiply eh term in the seond rket y eh term in the first, then dd. This mens you epnd out ( + d) + ( + d) to otin + d + + d. We use this ide gin when multiplying out inomil produts involving surds. Emple 9 Epnd nd simplify: ( 7 + )(5 7 4) (5 )( 4) ( 4 )(5 ) d ( )( + ) ( 7 + )(5 7 4) 7(5 7 4) + (5 7 4) 5 49 7 + 5 7 4 05 7 7 4 0 7 7 (5 )( 4) 5 ( 4) ( 4) 0 0 6 + 6 ( 4 )(5 ) (5 ) 4 (5 ) 5 6 6 60 + 4 6 9 6 66 d ( )( + ) + 4 50 IE-EM MTHEMTIS YER 9 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
D Multiplition nd division of surds Multiplition nd division of surds Emple 5 Emple 6 Emple 7 Emple 8 Emple 9 For positive numers nd d, d d. For positive numers nd d, d d, provided 0. Give the nswers in simplest form. We epnd inomil produts involving surds just s we do in lger: Simplify: ( + )( + d) ( + d) + ( + d) + d + + d Eerise D 5 7 d 4 7 e f 4 5 g 5 7 h 6 4 7 Simplify: 5 5 8 0 0 6 d e 4 5 5 f g 5 7 h i 6 5 4 j 4 40 7 5 k 0 0 6 l 7 8 4 7 Simplify: 6 6 6 7 7 d 4 5 0 e 6 f 6 g 7 0 h 4 7 4 Epnd nd simplify: ( 6 + ) ( 6) 5( 5) d ( ) e 5 5(4 ) f 7( 7 4) g 4 5( 5 ) h ( 5) i ( + 4 ) j 6 5( + ) k 7( 4) l 5( 5 + ) 5 Epnd nd simplify: (4 5 + )( 5 + ) (4 + 6)( + 5 6) ( + )( ) d ( + 5)(7 6 5) e ( 4)( + 5) f ( 7 )(5 7 ) g (7 + 5) h (4 ) HPTER Pythgors theorem nd surds 5 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
6 Epnd nd simplify: ( 5 + )( + ) ( )( + ) (4 + )( 7 5) d (8 4 6)( 7) e ( 7 5)( 7 + 5) f (5 )( + ) g (4 7 5)( 5 + 7) h (4 5)( 5 + ) 7 If nd y +, find: y + y + ( ) d y e y f + y g ( + )( y + ) h 6 0 + i + j + k + + E Speil produts In lger, you lerned the following speil epnsions. They re lled identities euse they re true for ll vlues of the pronumerls. These re espeilly importnt when deling with surds. ( + ) + + Emple 0 Epnd nd simplify ( 5 + ). ( ) + ( )( + ) ( 5 + ) ( 5) + 5 + ( ) 5 + 5 + 8 + 5 The first two identities re lled the perfet squre identities. The lst identity is known s the differene of two squres identity. You will need to e very onfident with these identities nd e le to reognise them. Sine they re true for ll numers, we n pply them to surds. Rell tht ( ) holds for ny positive numer. 5 IE-EM MTHEMTIS YER 9 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
E Speil produts Emple Epnd nd simplify ( ). ( ) ( ) + ( ) 8 6 + 0 6 You should lwys epress your nswer in simplest form. Emple Epnd nd simplify ( + 4 6). ( + 4 6) ( ) + 4 6 + (4 6) + 6 8 + 96 08 + 48 The following emple shows n interesting pplition of the differene of two squres identity. We will use this lter in this hpter when simplifying surds with inomil denominter. Emple Epnd nd simplify: ( 7 5)( 7 + 5) (5 6 5)(5 6 + 5) ( 7 5)( 7 + 5) ( 7) 5 7 5 8 (5 6 5)(5 6 + 5) (5 6) ( 5) 5 6 4 5 0 HPTER Pythgors theorem nd surds 5 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
E Speil produts Identities The following identities re often used in lultions involving surds. ( + ) + + ( ) + ( )( + ) Emple 0 Emple Emple Emple Eerise E Epnd nd simplify: (5 + ) ( + 6) (4 + 5) d ( + ) e ( 5 + 7) f ( + ) g (5 + ) h ( + 5) i ( 5 + 4 ) j ( + y) k ( + y) l ( y + ) Epnd nd simplify: ( 7 ) (4 ) ( 5 ) d ( ) e ( 5 ) f ( ) g ( 4 5) h (4 7) i ( y) Epnd nd simplify: ( 0 + 4 5) (5 6 + ) ( + ) d ( 5 + 5) e ( 0 ) f ( 0) g ( 70 0) h ( 50 5) i ( ) j ( 0 5) k ( 6) l (5 4 ) 4 Epnd nd simplify: ( 5)( + 5) ( 6 )( 6 + ) (7 + )(7 ) d ( )( + ) e ( 5 + )( 5 ) f ( )( + ) g ( 5)( + 5) h (4 5)(4 + 5) i (6 7)(6 + 7) j ( 5 6)( + 5 6) k ( 5 + 7 0)( 5 7 0) l ( + y)( y) 5 If nd y +, find: y + y d y e y f y g y h + y 54 IE-EM MTHEMTIS YER 9 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
6 Find: ( 5 + ) + ( 5 ) ( 5 + ) ( 5 ) ( + ) + ( ) d ( + ) ( ) e ( 7 ) + ( 7 + ) f ( 6 + ) ( 6 ) 7 Simplify ( + d)( d). 8 For the figures shown, find: i the vlue of ii the re of the tringle iii the perimeter of the tringle + 9 For the figures shown, find: i the perimeter 6 6 + ii the re 4 + 4 + 0 n equilterl tringle hs sides length +. Find: the perimeter of the tringle the re of the tringle F Rtionlising the denomintor 5 4 5 + The epression + is untidy. The first term hs squre root in the denomintor. Frtions involving surds re esiest to del with when they re epressed in simplest form, with rtionl denomintor. When we multiply the numertor nd denomintor of frtion y the sme numer, we form n equivlent frtion. The sme hppens with quotient involving surds. HPTER Pythgors theorem nd surds 55 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
F Rtionlising the denomintor Emple 4 Epress 4 with rtionl denomintor. 4 4 4 Emple 5 9 4 9 Write s quotient with rtionl denomintor. 4 9 4 9 4 Emple 6 Simplify +. + + + 4 + 6 6 7 6 (Multiply top nd ottom y the sme surd.) (Multiply top nd ottom y.) (Rtionlise the denomintor of the first term.) (Use ommon denomintor.) 56 IE-EM MTHEMTIS YER 9 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
F Rtionlising the denomintor Rtionlising the denomintor Rtionlising the denomintor mens onverting the denomintor into rtionl numer. We use the result, where is positive, to rtionlise the denomintor of quotient involving surds. Emple 4 Emple 5 Emple 6 Rtionlise the denomintor. 5 5 6 f Eerise F 8 6 g 5 Rtionlise the denomintor. 7 5 f 5 g 4 5 7 h 6 4 7 h 8 8 d 4 7 i d i 4 7 5 7 4 6 e 7 j 7 e j 0 5 5 Given tht.4 nd.7, find these vlues, orret to deiml ples, without using lultor. (First rtionlise the denomintor where pproprite.) 5 d e f 8 g h 8 4 y first rtionlising the denomintors, simplify eh epression. + + 5 d 4 5 5 e 7 + f + + 7 5 If 4 nd y 4, find nd rtionlise the denomintor. y y y d y HPTER Pythgors theorem nd surds 57 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
6 If, find nd rtionlise the denomintor. + + d 7 Find the vlue of. Epress your nswer with rtionl denomintor. 5 G 0 pplitions of Pythgors theorem in three dimensions How n we find the length of the digonl D of ue whose side length is 4 m? D is fe digonl nd sometimes D is lled spe digonl. We n pply Pythgors theorem to tringle D to find the squre of the length s m of the digonl D. s 4 + 4 We n then pply Pythgors theorem gin to tringle D sine D is right ngle. The length d m of the digonl D is given y: d s + 4 + 6 48 d 48 4 6.9 (orret to deiml ples) The length of the digonl D is 6.9 m orret to deiml ples. 4 m 5 4 m d m s m D 58 IE-EM MTHEMTIS YER 9 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
G pplitions of Pythgors theorem in three dimensions Emple 7 squre pyrmid hs height 5m nd squre se with side length 4m. Find the length of the edge V in this digrm. V 5 m m The se is squre with side length 4m, so: 4 + 4 4 Hene E (Leve in et form to mintin ury.) Tringle VE is right-ngled, so: VE + E 5 + ( ) 5 + 8 5.74 (orret to deiml ples) The lenth of V is 5.74 m orret to deiml ples. pplitions of Pythgors theorem in three dimensions Pythgors theorem n e used to find lengths in three-dimensionl prolems. lwys drw reful digrm identifying the pproprite right-ngled tringle(s). To mintin ury, use et vlues nd only pproimte using lultor t the end of the prolem if required. 4 m E D 4 m HPTER Pythgors theorem nd surds 59 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
G pplitions of Pythgors theorem in three dimensions Eerise G Emple 7 The retngulr prism in the digrm hs length of m, width of 5 m nd height of 6m. onsider tringle EFG. Find: i EF Find EG. Find G, orret to deiml ple. ii the size of ÐEFG Note: G is lled the spe digonl of the retngulr prism. Find the length of the spe digonl of the retngulr prism whose length, width nd height re: m,9 m, 8m m,5m,8m 0 m,4 m,7 m d 8m, 6m, 4m e 7m, m, m f m, m, m Find the length of the longest penil tht n fit inside ylindril penil se of length 5 m nd rdius m. 4 owl in the shpe of hemisphere of rdius length 5m is prtilly filled with wter. The surfe of the wter is irle of rdius 4 m when the rim of the owl is horizontl. Find the depth of the wter. 5 uilder needs to rry lengths of timer long orridor in order to get them to where he is working. There is right-ngled end in the orridor long the wy. The orridor is m wide nd the eiling is.6 m ove the floor. Wht is the longest length of timer tht the uilder n tke round the orner in the orridor? (Hint: Drw digrm.) 6 oin for n industril knitting mhine is in the shpe of trunted one. The dimeter of the top is 4m, the dimeter of the se is 6m nd the length of the slnt is 0 m. Find the height of the oin. 7 For the retngulr prism shown opposite, EH 4 m nd HG m. Find the et length of EG, giving your nswer s surd in simplest form. If E EG, find the et vlue of E. E F D H E 4 m 6 m m D G 0 m H G 6 m F 5 m 60 IE-EM MTHEMTIS YER 9 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
Find the length of: i E ii H d Wht type of tringle is tringle EH? e Show tht if EH m, HG m nd E EG, then the sides of the tringle EH re in the rtio : 4 : 5. H inomil denomintors onsider the epression. 7 5 How n we write this s quotient with rtionl denomintor? In the setion on speil produts, we sw tht: so Similrly, so ( ) ( ) ( 7 5)( 7 + 5) 7 5 7 5 7 + 5 7 5 7 + 5 7 5 7 5, whih is rtionl, 7 + 5 7 + 5 (5 4)(5 + 4) (5 ) 4 5 4 4, whih gin isrtionl, 5 + 4 5 4 5 + 4 5 + 4 Using the differene of two squres identity in this wy is n importnt tehnique. HPTER Pythgors theorem nd surds 6 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
H inomil denomintors Emple 8 Simplify the following: 5 5 + + 5 5 5 5 + 5 5 + 0 + 4 5 0 4 0 + 4 5 6 4(5 + 5) 6 5 + 5 4 inomil denomintors + + + + ( + )( ) 8 ( 6 6 + 6 6) 6 6 6 To rtionlise denomintor whih hs two terms, we use the differene of two squres identity: In n epression suh s In n epression suh s Eerise H 5 +, multiply top nd ottom y 5. 7, multiply top nd ottom y 7 +. The surd 7 is lled the onjugte of 7 + nd 7 + is the onjugte of 7. Simplify: ( 7 5)( 7 + 5) ( 5 )( 5 + ) (7 + 4 )(7 4 ) d (4 )(4 + ) 6 IE-EM MTHEMTIS YER 9 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
Emple 8 Rtionlise the denomintor in eh epression. 6 + + 5 d 5 e 4 5 + f 7 5 g j 6 + 5 5 5 6 m 4 7 h k n 5 + 4 6 + 5 i l o 5 + 6 + 0 Rtionlise the denomintors in these epressions nd use the deiml pproimtions.44 nd.7 to evlute them orret to deiml ples. + 4 Find the integers p nd q suh tht 5 Simplify: + 5 6 Simplify: I 5 + 5 + 5 + 5 + 5 5 p + q 5. 5 5 5 + Irrtionl numers nd surds We first rell some fts out frtions nd deimls. Frtions nd deimls + 6 5 + 5 + d 5 ( 7 ) We know tht some frtions n e written s deimls tht terminte. For emple, 4 0.5, 6 0.875 Some frtions hve deiml representtions tht do not terminte. For emple, 0.... whih we write s 0... HPTER Pythgors theorem nd surds 6 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
I Irrtionl numers nd surds The dot ove the indites tht the digit is repeted forever. Some frtions hve deiml representtions tht eventully repet. For emple, 6 0.666... whih is written s 0.6. Other frtions hve deiml representtions with more thn one repeting digit. For emple, 0.090909... whih we write s 0.0. 9., with dot ove oth of the repeting digits. nother emple is 0.. 456. 0. 456 456... onverting deimls to frtions It is esy to write terminting deimls s frtions y using denomintor whih is power of 0. For emple, 4 7 0.4 00 50 Deimls tht hve repeted sequene of digits n lso e written s frtions. For emple, 4 0. nd 0.67 44 660 method for doing this is shown in the net two emples. Emple 9 Write 0.. s frtion. Let S 0.. So S 0.... Then 0S.... (Multiply y 0.) 0S + 0.... Therefore 0S S + Hene 9S So S 9 Thus 0.. 9 64 IE-EM MTHEMTIS YER 9 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
I Irrtionl numers nd surds Emple 0 Write 0.. s frtion. Let S 0.. So S 0.... Then 00S... Therefore 00S + S Hene 99S So S 99 4 4 Thus 0. Rtionl numers rtionl numer is numer tht n e written s frtion p q, where p nd q re integers nd q 0. ll integers re rtionl numers. For emple, Emple Eplin why eh of these numers is rtionl. 5 7 We will epress eh numer in the form p q..7 0.4 8 5 7 7.7 7 0 000 7 0 000 (ontinued over pge) HPTER Pythgors theorem nd surds 65 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
I Irrtionl numers nd surds Let S 0.4 0.4 4... 00S 4.4 4... (Multipyy00.) 00S 4 + S Hene 99S 4 4 S 99 8 8 Thus 0.4 Eh numer hs een epressed s frtion p, where p nd q re integers, so eh numer is rtionl. q Irrtionl numers Mthemtiins up to out 600 E thought tht ll numers were rtionl. However, when we pply Pythgors theorem, we enounter numers suh s tht re not rtionl. The numer is n emple of n irrtionl numer. n irrtionl numer is one tht is not rtionl. Hene n irrtionl numer nnot e written in the p form, where p nd q re integers nd q 0. Nor n it e written s terminting or repeting q deiml. In 00 E, Eulid proved tht mking the ssumption tht is rtionl leds to ontrdition. Hene ws shown to e irrtionl. The proof is outlined here. ssume p where p nd q re integers with highest ommon ftor. q p Squre oth sides of this eqution to otin q. We n write p q. Hene p is even nd thus p is even. We n now write, 4 k q For some whole numer k. From this, show q is lso even whih is ontrdition of our ssumption tht the highest ommon ftor is. The deiml epnsion of goes on forever ut does not repet. The vlue of n e pproimted using lultor. The sme is true of other irrtionl numers suh s, 4 nd 9. Try finding these numers on your lultor nd see wht you get. Other emples of irrtionl numers inlude:,, ππ, nd π There re infinitely mny rtionl numers euse every whole numer is rtionl. We n lso esily write down infinitely mny other frtions. For emple,,, 4,... 66 IE-EM MTHEMTIS YER 9 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
I Irrtionl numers nd surds There re infinitely mny irrtionl numers too. For emple,,,, 4,... Every numer, whether rtionl or irrtionl, is represented y point on the numer line. onversely, we n think of eh point on the numer line s numer. π 4 0 4 Emple rrnge these irrtionl numers in order of size on the numer line. 4 8 60 0 Find on pproimtion of eh numer to deiml ples using lultor. 8.8.4 4 60.9 0.4 0 Surds 4 0 8 In Yer 8, we looked t squres, squre roots, ues nd ue roots. For emple: 5 5 nd 5 5 5 5 nd 5 5 60 The use of this nottion n e etended. For emple: 5 4 65 nd 65 5 4 This is red s The fourth power of 5 is 65 nd the fourth root of 65 is 5. We lso hve fifth roots, sith roots nd so on. In generl, we n tke the nth root of ny positive numer. The nth root of is the positive numer whose nth power is. The sttement n is equivlent to the sttement n. Your lultor will give you pproimtions to the nth root of. Note: For n, positive integer, 0n n 0 nd 0 0. n n irrtionl numer whih n e epressed s, where is positive whole numer, is lled surd. For emple,, 7 nd 5 re ll emples of surds, while 4 nd 9 re not surds sine they re whole numers. The numer π, lthough it is irrtionl, is not surd. This is lso diffiult to prove. 4 HPTER Pythgors theorem nd surds 67 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
I Irrtionl numers nd surds Emple Use Pythgors theorem to onstrut line of length 0. Emple 9 Emple 0 Emple Find two perfet squres tht sum to 0. 4 + 0 Drw perpendiulr line segments from ommon point of lengths units nd 4 units. onnet their endpoints with third line segment. 4 0 Irrtionl numers nd surds Every frtion n e written s terminting, or eventully repeting, deiml. Every terminting, or eventully repeting, deiml n e written s frtion. rtionl numer is numer whih n e epressed s p where p nd q re integers q nd q 0. Tht is, rtionl numer is n integer or frtion. There re numers suh s π nd Eerise I Write eh repeting deiml s frtion. tht re irrtionl (not rtionl). Eh rtionl nd eh irrtionl numer represents point on the numer line. Every point on the numer line represents rtionl or n irrtionl numer. n irrtionl numer whih n e epressed s numers, is lled surd. π is not surd., where n nd re positive whole 0.5 0.7 0.9 d 0.4 e 0. f 0.6 Write eh repeting deiml s frtion. 0. 0.07 0.9 d 0.4 e 0.6 f 0.06 g 0.4 h 0.56 i 0.00 Show tht eh numer is rtionl y writing it s frtion. 5.5 0.4 d 0.5 e 5 7 f. n 68 IE-EM MTHEMTIS YER 9 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
Emple Emple 4 Whih of these numers re irrtionl? 7 5 0.6 d π 5 y pproimting orret to deiml ples, ple these rel numers on the sme numer line. 7.7 8 d π e 6 Whih of these numers re surds? 5 4 d 9 e 7 f 6 g 0 h i j 5 k 5 l 5 7 Use the ft tht + 5 to onstrut length 5. Use the ft tht 4 + 5 4 to onstrut length 4. 8 How would you onstrut n intervl of length: 7??? 9 Show tht 8 7 + 9 + + 4 The epression on the right is lled ontinued frtion. Epress 5 s ontinued frtion with ll numertors. Review eerise For eh right-ngled tringle, find the vlue of the pronumerl. h m 8 m 6 m.5 m 6 m h m For eh right-ngled tringle, find the vlue of the pronumerl. m 4 m 8.5 m 6.5 m y m 6 m HPTER Pythgors theorem nd surds 69 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
REVIEW EXERISE support rket is to e pled under shelf, s shown in the digrm. If 0 m, find, orret to the nerest millimetre, the length of. 4 rod runs in n est west diretion joining towns nd, whih re 40 km prt. third town,, is situted 0 km due north of. stright rod is uilt from, to the rod etween nd nd meets it t D, whih is equidistnt from nd. Find the length of rod D. 5 irles of rdius 6 m nd m re pled in squre, s shown in the digrm opposite. Find, orret to deiml ples: E FD the length of the digonl D d the side length of the squre 6 Simplify: + 8 7 Simplify: 5 6 5 0 8 Simplify: ( + ) 5 ( ) 9 Epnd nd simplify: ( + )( ) (5 )( ) 0 Simplify: 80 08 5 d 7 e 048 f 448 g 800 h Simplify: 45 + 5 80 8 + 6 5 7 44 + 75 4 d 6 00 + 88 Epress with rtionl denomintor. 5 5 e f 5 5 4 7 7 g 4 7 D d F 7 h E 70 IE-EM MTHEMTIS YER 9 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
REVIEW EXERISE Epress with rtionl denomintor. + 5 0 e f 4 7 5 g 7 4 d + 7 0 h 4 Epnd nd simplify ( 5 ). Simplify +, epressing your nswer with rtionl denomintor. 5 The digrm opposite shows prt of skte ord rmp. (It is prism whose ross-setion is rightngled tringle.) Use the informtion in the digrm to find: E 6 Find the length of the spe digonl of ue with side length 5m. 7 motorist deprts from town, whih is 8km due south from nother town,, nd drives due est towrds town, whih is 0 km from. fter driving distne of km, he noties tht he is the sme distne wy from oth towns nd. 8 km km D 0 km Epress the motorist s distne from in terms of (tht is, D). Epress the motorist s distne from in terms of. Find the distne he hs driven from. 8 The digrm opposite shows the logo of prtiulr ompny. The lrge irle hs entre O nd rdius m nd is dimeter. D is the entre of the middle-sized irle with dimeter O. Finlly, is the entre of the smllest irle. Wht is the rdius of the irle with entre D? If the rdius of the irle with entre is r m, epress these in terms of r. i O ii D Find the vlue of r. D 5 m O 4 m D F.5 m HPTER Pythgors theorem nd surds 7 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
9 In the digrm opposite, VD is squre-sed pyrmid with D D 0 nd V V V VD 0. Wht type of tringle is tringlev? If M is the midpoint of, find the et vlues of: i VM ii VE, the height of the pyrmid 0 Write eh repeting deiml s frtion. V M E D 0.8 0.5 0.7 d 0.0 e 0.8 f 0.96 g 0.0 h 0.005 Write 67 s ontinued frtion. (See Eerise I, Question 9.) 9 Write 0.479 8 s frtion. How would you onstrut intervls of the following lengths? Disuss with your teher. 8 d 8 4 Find the length of the long digonl of the retngulr prism whose length, width nd height re: +,, 5 +, 5, 5 For nd y 5, find in simplest form. + + d y y y hllenge eerise Prove tht is irrtionl. In the tringle elow, 4 + 7 65 > 8. Is the ngle θ greter or less thn 90? Eplin your nswer. 7 8 θ 4 7 IE-EM MTHEMTIS YER 9 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
HLLENGE EXERISE pollonius theorem sttes tht in ny tringle, if we join verte to the midpoint of the opposite side, nd the length of tht line is d, then + ( d + m)where m. In words it sys: In ny tringle, the sum of the squres on two sides is equl to twie the squre on hlf the third side together with twie the squre on the medin. Prove this result s follows: Drop perpendiulr E of length h from to nd let ED p. i Write E in terms of m nd p. ii Write E in terms of m nd p. y pplying Pythgors theorem to the three right-ngled tringles in the digrm, omplete these sttements: d dd the first two equtions in prt ove nd simplify. d Use the third eqution in prt to dedue pollonius theorem. e Wht hppens when the ngle t is right ngle? m h 4 retngle D hs 5 m, D 0m. point P is loted inside the retngle suh tht P 4 m nd P m. Find PD in et form. 5 E is ny point inside the retngle D. Let E, E, E nd ED. Prove tht +. d D d E p D m m HPTER Pythgors theorem nd surds 7 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400
HLLENGE EXERISE 6 We use the ft tht every integer n e uniquely ftorised s produt of primes to give nother proof tht is irrtionl. We egin y supposing the opposite tht is, we suppose tht we n write p, where p nd q re whole numers with no q ommon ftors eept. Eplin why q is ftor of p. Eplin why q is ftor of p. Eplin how this shows tht is irrtionl. 7 Prove tht 6 is irrtionl. 8 Epnd nd simplify: ( + ) 4 ( ) 4 9 Prove tht there re infinitely mny irrtionl numers etween 0 nd. 0 Prove tht etween ny two numers, there re infinitely mny rtionl numers nd infinitely mny irrtionl numers. Rtionlise the denomintor of: ( + ) + 5 7 5 7 + Show tht 7 < 5 < < without using lultor. Two irles with entres O nd O nd rdii r nd R meet t single point. nd re the points where the irles touh line l. Find the distne etween points nd. O r O R 74 IE-EM MTHEMTIS YER 9 Finl pges mridge University Press rown et l, 07 ISN 978--08-404-7 Ph 0 867 400