# The theorem of. Pythagoras. Opening problem

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1 The theorem of 8 Pythgors ontents: Pythgors theorem [4.6] The onverse of Pythgors theorem [4.6] Prolem solving [4.6] D irle prolems [4.6, 4.7] E Three-dimensionl prolems [4.6] Opening prolem The Louvre Pyrmid in Pris, Frne hs squre se with edges 35 m long. The pyrmid is 20:6 m high. n you find the length of the slnt edges of the pyrmid? Right ngles ( 90 o ngles) re used when onstruting uildings nd dividing res of lnd into retngulr regions. The nient Egyptins used rope with 12 eqully sped knots to form tringle with sides in the rtio 3:4:5: This tringle hs right ngle etween the sides of length 3 nd 4 units. In ft, this is the simplest right ngled tringle with sides of integer length.

2 170 The theorem of Pythgors (hpter 8) The Egyptins used this proedure to onstrut their right ngles: orner tke hold of knots t rrows mke rope tut line of one side of uilding PYTHGORS THEOREM [4.6] right ngled tringle is tringle whih hs right ngle s one of its ngles. The side opposite the right ngle is lled the hypotenuse nd is the longest side of the tringle. The other two sides re lled the legs of the tringle. round 500, the Greek mthemtiin Pythgors disovered rule whih onnets the lengths of the sides of ll right ngled tringles. It is thought tht he disovered the rule while studying tesselltions of tiles on throom floors. Suh ptterns, like the one illustrted, were ommon on the wlls nd floors of throoms in nient Greee. hypotenuse legs PYTHGORS THEOREM In right ngled tringle with hypotenuse nd legs nd, 2 = y looking t the tile pttern ove, n you see how Pythgors my hve disovered the rule? In geometri form, Pythgors theorem is: In ny right ngled tringle, the re of the squre on the hypotenuse is equl to the sum of the res of the squres on the other two sides. X X X GEOMETRY PKGE

3 The theorem of Pythgors (hpter 8) 171 There re over 400 different proofs of Pythgors theorem. Here is one of them: On squre we drw 4 identil (ongruent) right ngled tringles, s illustrted. smller squre is formed in the entre. Suppose the legs re of length nd nd the hypotenuse hs length. The totl re of the lrge squre =4 re of one tringle + re of smller squre, ) ( + ) 2 =4( 1 2 )+2 ) =2 + 2 ) = 2 Exmple 1 Find the length of the hypotenuse in: The hypotenuse is opposite the right ngle nd hs length. ) x 2 = ) x 2 =9+4 ) x 2 =13 ) x = p 13 fs x>0g 2m ) the hypotenuse is out 3:61 m long. If x 2 = k, then x = p k, ut we rejet p k s lengths must e positive. Exmple 2 Find the length of the third side of this tringle: 6m 5m The hypotenuse hs length 6 m. ) x =6 2 fpythgorsg ) x 2 +25=36 ) x 2 =11 ) x = p 11 fs x>0g ) the third side is out 3:32 m long.

4 172 The theorem of Pythgors (hpter 8) Exmple 3 Find x in surd form: ~`1`0 m 2m x m 2x m 6m The hypotenuse hs length. ) x 2 =2 2 +( p 10) 2 fpythgorsg ) x 2 =4+10 ) x 2 =14 ) x = p 14 fs x>0g (2x) 2 = x fpythgorsg ) 4x 2 = x ) 3x 2 =36 ) x 2 =12 ) x = p 12 fs x>0g ) x =2 p 3 Exmple 4 Find the vlue of y, giving your nswer orret to 3 signifint figures. y m 5m 1m D 6m In tringle, the hypotenuse is. ) x 2 = fpythgorsg ) x 2 =26 ) x = p 26 fs x>0g In tringle D, the hypotenuse is 6 m. Sine we must find the vlue of y, we leve x in surd form. Rounding it will redue the ury of our vlue for y. ) y 2 +( p 26) 2 =6 2 fpythgorsg ) y 2 +26=36 ) y 2 =10 ) y = p 10 fs y>0g ) y ¼ 3:16 EXERISE Find the length of the hypotenuse in the following tringles, giving your nswers orret to 3 signifint figures: 4m x km 7m 8km 5m 13 km

5 The theorem of Pythgors (hpter 8) Find the length of the third side these tringles, giving your nswers orret to 3 signifint figures: x km 6m 11 m 2.8 km 3 Find x in the following, giving your nswers in simplest surd form: 4 Solve for x, giving your nswers in simplest surd form: Qw_ m 5 Find the vlues of x, giving your nswers orret to 3 signifint figures: 9m 2x m 1m ~`2 m Qw_ m ~`7 m 26 m ~`5 m Ew_ m 3x m 6 Find the vlue of ny unknowns, giving nswers in surd form: 1.9 km 2x m 2m 9.5 m ~`1`0 m 2x m 3x m 2 1m m 1m x m ~`2`0 m y m 4m y m 7m y m 2m 2m 7 Find x, orret to 3 signifint figures: ( x ) m 4m 5m 13 m 8 Find the length of side orret to 3 signifint figures: 5m 9m D

6 174 The theorem of Pythgors (hpter 8) 9 Find the distne in the following: D 4m 4m M 6m 7m 3m hllenge 5m 1m N 10 In 1876, President Grfield of the US pulished proof of the theorem of Pythgors. longside is the figure he used. Write out the proof. E D 11 You re given retngle in whih there is point tht is 3 m, 4 m nd 5 m from three of the verties. How fr is the point from the fourth vertex? 4m 5m PYTHGOREN TRIPLES The simplest right ngled tringle with sides of integer length is the tringle. 5 3 The numers 3, 4, nd 5 stisfy the rule = The set of positive integers f,, g is Pythgoren triple if it oeys the rule = 2 : Other exmples re: f5, 12, 13g, f7, 24, 25g, f8, 15, 17g. Exmple 5 Show tht f5, 12, 13g is Pythgoren triple. We find the squre of the lrgest numer first = 169 nd = = 169 ) =13 2 So, f5, 12, 13g is Pythgoren triple.

7 #endoxedheding The theorem of Pythgors (hpter 8) 175 Exmple 6 Find k if f9, k, 15g is Pythgoren triple. Let k 2 =15 2 fpythgorsg ) 81 + k 2 = 225 ) k 2 = 144 ) k = p 144 fs k>0g ) k =12 EXERISE Determine if the following re Pythgoren triples: f8, 15, 17g f6, 8, 10g f5, 6, 7g d f14, 48, 50g e f1, 2, 3g f f20, 48, 52g 2 Find k if the following re Pythgoren triples: f8, 15, kg fk, 24, 26g f14, k, 50g d f15, 20, kg e fk, 45, 51g f f11, k, 61g 3 Explin why there re infinitely mny Pythgoren triples of the form f3k, 4k, 5kg where k 2 Z +. Disovery Pythgoren triples spredsheet Well known Pythgoren triples inlude f3, 4, 5g, f5, 12, 13g, f7, 24, 25g nd f8, 15, 17g. SPREDSHEET Formule n e used to generte Pythgoren triples. n exmple is 2n +1, 2n 2 +2n, 2n 2 +2n +1 where n is positive integer. spredsheet n quikly generte sets of Pythgoren triples using suh formule. Wht to do: 1 Open new spredsheet nd enter the following: in olumn, the vlues of n for n =1, 2, 3, 4, 5,... in olumn, the vlues of 2n +1 in olumn, the vlues of 2n 2 +2n d in olumn D, the vlues of 2n 2 +2n +1. fill down 2 Highlight the pproprite formule nd fill down to Row 11 to generte the first 10 sets of triples. 3 hek tht eh set of numers is indeed triple y dding olumns to find nd 2. 4 Your finl tsk is to prove tht the formule f2n +1, 2n 2 +2n, 2n 2 +2n +1g will produe sets of Pythgoren triples for ll positive integer vlues of n. Hint: Let = 2n +1, = 2n 2 +2n nd = 2n 2 +2n +1, then simplify 2 2 =(2n 2 +2n +1) 2 (2n 2 +2n) 2 using the differene of two squres ftoristion.

8 176 The theorem of Pythgors (hpter 8) THE ONVERSE OF PYTHGORS THEOREM [4.6] If we hve tringle whose three sides hve known lengths, we n use the onverse of Pythgors theorem to test whether it is right ngled. THE ONVERSE OF PYTHGORS THEOREM If tringle hs sides of length, nd units nd = 2, then the tringle is right ngled. GEOMETRY PKGE Exmple 7 Is the tringle with sides 6 m, 8 m nd 5 m right ngled? The two shorter sides hve lengths 5 m nd 6 m. Now = = 61, ut 8 2 =64: ) =8 2 nd hene the tringle is not right ngled. EXERISE 8 1 The following figures re not drwn to sle. Whih of the tringles re right ngled? 7m 9m 12 m 5m 5m 4m 15 m 9m 8m d e f ~`7 m ~` `7 m ~`4`8 m ~`1`2 m ~`7`5 m 15 m 8m 17 m 2 The following tringles re not drwn to sle. If ny of them is right ngled, find the right ngle. 2 m 1m 8m ~`2`0`8 m ~`2`4 km 5km ~`5 m 12 m 7km

9 The theorem of Pythgors (hpter 8) Ted hs two plnks 6800 mm long, nd two 6800 mm plnks 3500 mm long. He lys them down s orders for the onrete floor of his new grge. To hek tht the shpe is retngulr, Ted mesures digonl length. He finds it to e 3500 mm 3500 mm 7648 mm. Is Ted s floor retngulr? 6800 mm 4 Tringle hs ltitude N whih is 6 m long. N =9m nd N =4m. Is tringle right ngled t? 6m 9m N 4m PROLEM SOLVING [4.6] Mny prtil prolems involve tringles. We n pply Pythgors theorem to ny tringle tht is right ngled, or use the onverse of the theorem to test whether right ngle exists. SPEIL GEOMETRIL FIGURES The following speil figures ontin right ngled tringles: digonl In retngle, right ngles exist etween djent sides. onstrut digonl to form right ngled tringle. retngle In squre nd rhomus, the digonls iset eh other t right ngles. squre rhomus In n isoseles tringle nd n equilterl tringle, the ltitude isets the se t right ngles. isoseles tringle equilterl tringle Things to rememer ² Drw net, ler digrm of the sitution. ² Mrk on known lengths nd right ngles. ² Use symol suh s x to represent the unknown length. ² Write down Pythgors theorem for the given informtion. ² Solve the eqution. ² Where neessry, write your nswer in sentene form.

10 178 The theorem of Pythgors (hpter 8) Exmple 8 retngulr gte is 3 m wide nd hs 3:5 m digonl. How high is the gte? Let x m e the height of the gte. Now (3:5) 2 = x fpythgorsg ) 12:25 = x 2 +9 ) 3:25 = x 2 ) x = p 3:25 fs x>0g ) x ¼ 1:80 Thus the gte is pproximtely 1:80 m high. 3m 3.5 m x m Exmple 9 rhomus hs digonls of length 6 m nd 8 m. Find the length of its sides. 4m The digonls of rhomus iset t right ngles. Let eh side of the rhomus hve length. ) x 2 = fpythgorsg ) x 2 =25 ) x = p 25 fs x>0g ) x =5 Thus the sides re 5 m in length. Exmple 10 Two towns nd re illustrted on grid whih hs grid lines 5 km prt. How fr is it from to? 5km 15 km 10 km 2 = fpythgorsg ) 2 = = 325 ) = p 325 fs > 0g ) ¼ 18:0 So, nd re out 18:0 km prt.

11 The theorem of Pythgors (hpter 8) 179 Exmple 11 n equilterl tringle hs sides of length 6 m. Find its re. The ltitude isets the se t right ngles. ) =6 2 fpythgorsg ) 2 +9=36 ) 2 =27 ) = p 27 fs >0g m 6m Now, re = 1 2 se height = p 27 =3 p 27 m 2 ¼ 15:6 m 2 So, the re is out 15:6 m 2 : Exmple 12 heliopter trvels from se sttion S for 112 km to outpost. It then turns 90 to the right nd trvels 134 km to outpost. How fr is outpost from se sttion S? o Let S e x km. From the digrm longside, we see in tringle S tht S =90 o. x 2 = fpythgorsg ) x 2 = ) x = p fs x>0g ) x ¼ 175 So, outpost is 175 km from se sttion S. S 112 km x km 134 km EXERISE 8 1 retngle hs sides of length 8 m nd 3 m. Find the length of its digonls. 2 The longer side of retngle is three times the length of the shorter side. If the length of the digonl is 10 m, find the dimensions of the retngle. 3 retngle with digonls of length 20 m hs sides in the rtio 2:1. Find the: perimeter re of the retngle. 4 rhomus hs sides of length 6 m. One of its digonls is 10 m long. Find the length of the other digonl. 5 squre hs digonls of length 10 m. Find the length of its sides.

12 180 The theorem of Pythgors (hpter 8) 6 rhomus hs digonls of length 8 m nd 10 m. Find its perimeter. 7 On the grid there re four towns,, nd D. The grid lines re 5 km prt. How fr is it from: to to tod d Dto e to f tod? Give ll nswers orret to 3 signifint figures. D 8 yht sils 5 km due west nd then 8 km due south. How fr is it from its strting point? 9 2km 4km 10 km ylist t is trvelling towrds. How fr will he hve to yle efore he is equidistnt from nd? 10 street is 8 m wide, nd there re street lights positioned either side of the street every 20 m. How fr is street light X from street light: d D? D X 11 Find ny unknowns in the following: 45 2m y 7m h m 12 m 1m y m 12 n equilterl tringle hs sides of length 12 m. Find the length of one of its ltitudes. 13 The re of tringle is given y the formul = 1 2 h: n isoseles tringle hs equl sides of length 8 m nd se of length 6 m. Find the re of the tringle. n equilterl tringle hs re 16 p 3 m 2. Find the length of its sides. h 8m 6m 14 string 1m 10 m string Hether wnts to hng 7 m long nner from the roof of her shop. The hooks for the strings re 10 m prt, nd Hether wnts the top of the nner to hng 1 m elow the roof. How long should eh of the strings e? 7m

13 The theorem of Pythgors (hpter 8) Two ushwlkers set off from se mp t the sme time, wlking t right ngles to one nother. One wlks t n verge speed of 5 km/h, nd the other t n verge speed of 4 km/h. Find their distne prt fter 3 hours. 16 To get to shool from her house, Ell wlks down ernrd Street, then turns 90 o nd wlks down Thompson Rod until she rehes her shool gte. She wlks twie s fr long ernrd Street s she does long Thompson Rod. If Ell s house is 2:5 km in stright line from her shool gte, how fr does Ell wlk long ernrd Street? 17 ot is 10 km est of ot. ot trvels 6 km north, nd ot trvels 2 km west. How fr prt re the ots now? D IRLE PROLEMS [4.6, 4.7] There re ertin properties of irles whih involve right ngles. In these situtions we n pply Pythgors theorem. The properties will e exmined in more detil in hpter 27. NGLE IN SEMI-IRLE The ngle in semi-irle is right ngle. No mtter where is pled on the r, is lwys right ngle. O Exmple 13 irle hs dimeter XY of length 13 m. Z is point on the irle suh tht XZ is 5 m. Find the length YZ. From the ngle in semi-irle theorem, we know X ZY is right ngle. Let the length YZ e. ) x 2 =13 2 Z fpythgorsg ) x 2 = = 144 ) x = p 5m O 144 fs x>0g X 13 m ) x =12 So, YZ hs length 12 m. Y HORD OF IRLE The line drwn from the entre of irle t right ngles to hord isets the hord. This follows from the isoseles tringle theorem. The onstrution of rdii from the entre of the irle to the end points of the hord produes two right ngled tringles. entre O rdius hord

14 182 The theorem of Pythgors (hpter 8) Exmple 14 irle hs hord of length 10 m. If the rdius of the irle is 8 m, find the shortest distne from the entre of the irle to the hord. The shortest distne is the perpendiulr distne. The line drwn from the entre of irle, perpendiulr to hord, isets the hord, so = =5m: In O, x 2 =8 2 fpythgorsg ) x 2 =64 25 = 39 ) x = p 39 fs x>0g ) x ¼ 6:24 8m 5m O 10 m So, the shortest distne is out 6:24 m. TNGENT-RDIUS PROPERTY tngent to irle nd rdius t the point of ontt meet t right ngles. Notie tht we n now form right ngled tringle. entre O rdius tngent point of ontt Exmple 15 tngent of length 10 m is drwn to irle with rdius 7 m. How fr is the entre of the irle from the end point of the tngent? Let the distne e d m. ) d 2 = fpythgorsg ) d 2 = 149 ) d = p 149 fs d>0g ) d ¼ 12:2 So, the entre is 12:2 m from the end point of the tngent. 7m O 10 m d m Exmple 16 Two irles hve ommon tngent with points of ontt t nd. The rdii re 4 m nd 2 m respetively. Find the distne etween the entres given tht is 7 m.

15 The theorem of Pythgors (hpter 8) 183 2m E 2m D 7m 7m 2m For entres nd D, we drw, D, D nd E k. ) E is retngle ) the distne etween the entres is out 7:28 m. ) E =7m fs E = g nd DE =4 2=2m Now x 2 = fpythgors in DEg ) x 2 =53 ) x = p 53 fs x>0g ) x ¼ 7:28 EXERISE 8D T 1 T is tngent to irle with entre O. The irle hs rdius 5 m nd =7m. Find the length of the tngent. 5m O 2 irle hs entre O nd rdius of 8 m. hord is O 13 m long. Find the shortest distne from the hord to the entre of the irle. 3 is dimeter of irle nd is hlf the length of. If is 12 m long, wht is the rdius of the irle? O 4 retngle with side lengths 11 m nd 6 m is insried in irle. Find 11 m the rdius of the irle. 6m 5 irle hs dimeter of length 10 m. is point on the irle suh tht is 8 m. Find the length. 6 squre is insried in irle of rdius 6 m. Find the length of the sides of the squre, orret to 3 signifint figures. 6m

16 184 The theorem of Pythgors (hpter 8) 7 hord of irle hs length 3 m. If the irle hs rdius 4 m, find the shortest distne from the entre of the irle to the hord. 8 hord of length 6 m is 3 m from the entre of irle. Find the length of the irle s rdius. 9 hord is 5 m from the entre of irle of rdius 8 m. Find the length of the hord. 10 irle hs rdius 3 m. tngent is drwn to the irle from point P whih is 9 m from O, the irle s entre. How long is the tngent? Leve your nswer in surd form. 11 Find the rdius of irle if tngent of length 12 m hs its end point 16 m from the irle s entre. 12 Two irulr pltes of rdius 15 m re pled in opposite orners of retngulr tle s shown. Find the distne etween the entres of the pltes. 80 m 1.5 m m nd re the entres of two irles with rdii 4 m nd 3 m respetively. The illustrted ommon tngent hs length 10 m. Find the distne etween the entres orret to 2 deiml ples. 14 Two irles re drwn so they do not interset. The lrger irle hs rdius 6 m. ommon tngent is 10 m long nd the entres re 11 m prt. Find the rdius of the smller irle, orret to 3 signifint figures. 10 m 15 The following figures hve not een drwn to sle, ut the informtion mrked on them is orret. Wht n you dedue from eh figure? O 4m 2m P 1.56 m 1.69 m Q 0.65 m R 16 ny two irles whih do not interset hve two ommon externl tngents s illustrted. The lrger irle hs rdius nd the smller one hs rdius. The irles re 2 units prt. Show tht eh ommon tngent hs length p 8( + ) units.

17 The theorem of Pythgors (hpter 8) hord of length 2 m is drwn in irle of rdius 3 m. dimeter is onstruted, nd the tngent from is drwn. The hord is extended to meet the tngent t D. Find the length of D. D 2m E THREE-DIMENSIONL PROLEMS [4.6] Pythgors theorem is often used to find lengths in three-dimensionl prolems. In these prolems we sometimes need to pply it twie. Exmple m rope is tthed inside n empty ylindril whet silo of dimeter 12 m s shown. How high is the whet silo? 12 m 50 m 50 m 12 m h m Let the height e h m. ) h =50 2 fpythgorsg ) h = 2500 ) h 2 = 2356 ) h = p 2356 fs h>0g ) h ¼ 48:5 So, the whet silo is pproximtely 48:5 m high. Exmple 18 The floor of room is 6 my4 m, nd its height is 3 m. Find the distne from orner point on the floor to the opposite orner point on the eiling. The required distne is D. We join D. In D, x 2 = fpythgorsg In D, y 2 = x fpythgorsg ) y 2 = ) y 2 =61 ) y = p 61 fs y>0g ) y ¼ 7:81 3m 4m x m 6m y m D ) the distne is out 7:81 m.

18 186 The theorem of Pythgors (hpter 8) Exmple 19 pyrmid of height 40 m hs squre se with edges 50 m. Determine the length of the slnt edges. 40 m x m s m 50 m Let slnt edge hve length s m. Let hlf digonl hve length x m. Using x 2 + x 2 =50 2 fpythgorsg x m x m ) 2x 2 = m ) x 2 = 1250 Using s 2 = x fpythgorsg ) s 2 = s m 40 m ) s 2 = 2850 ) s = p 2850 fs s>0g ) s ¼ 53:4 x m So, eh slnt edge is pproximtely 53:4 m long. EXERISE 8E 1 one hs slnt height of 17 m nd se rdius of 8 m. How high is the one? 2 ylindril drinking glss hs rdius 3 m nd height 10 m. n 12 m long thin stirrer e pled in the glss so tht it will sty entirely within the glss? 3 20 m nil just fits inside ylindril n. Three identil spheril lls need to fit entirely within the n. Wht is the mximum rdius of eh ll? 4 ui die hs sides of length 2 m. Find the distne etween opposite orners of the die. Leve your nswer in surd form. 5 room is 5 my3 m nd hs height of 3:5 m. Find the distne from orner point on the floor to the opposite orner of the eiling. 6 Determine the length of the longest metl rod whih ould e stored in retngulr ox 20 m y 50 m y 30 m. 7 tree is 8 m north nd 6 m est of nother tree. One of the trees is 12 m tll, nd the other tree is 17 m tll. Find the distne etween: the trunks of the trees the tops of the trees. 8 rinwter tnk is ylindril with onil top. The slnt height of the top is 5 m, nd the height of the ylinder is 9 m. Find the distne etween P nd Q, to the nerest m. 2m Q 5m 9m P 8m

19 #endoxedheding The theorem of Pythgors (hpter 8) my18 my4m hll is to e deorted with stremers for prty. 4m 4 stremers re tthed to the orners of the floor, nd 4 stremers re tthed to the entres of the wlls s illustrted. ll 8 stremers re then tthed to the entre of the eiling. 18 m Find the totl length of stremers required. 6m 10 nswer the Opening Prolem on pge 169. Review set 8 1 Find the lengths of the unknown sides in the following tringle. Give your nswers orret to 3 signifint figures. 2m 4m 5m 7m 9m 2x m 2 Is the following tringle right ngled? Give evidene. 5 ~`1`1 3 Show tht f5, 11, 13g is not Pythgoren triple. 4 Find, orret to 3 signifint figures, the distne from: to to to. 4km 5 rhomus hs digonls of length 12 m nd 18 m. Find the length of its sides. 6 irle hs hord of length 10 m. The shortest distne from the irle s entre to the hord is 5 m. Find the rdius of the irle. 7 The digonl of ue is 10 m long. Find the length of the sides of the ue. digonl

20 #endoxedheding 188 The theorem of Pythgors (hpter 8) 8 Two irles hve the sme entre. The tngent drwn from point P on the smller irle uts the lrger irle t Q nd R. Show tht PQ nd PR re equl in length. R 9 Find x, orret to 3 signifint figures: Q P tngent 2x m O 9m 5m 10 m 10 rn hs the dimensions given. 1.5 m Find the shortest distne from to. 3m 2m 5m Review set 8 1 Find the vlue of x: 2x 5m 7 m x m 5m 5x 6m 2 Show tht the following tringle is right ngled nd stte whih vertex is the right ngle: 2 ~`2`9 5 3 Is tringle right ngled? Give evidene to support your nswer. 10 m 5m 4m

21 The theorem of Pythgors (hpter 8) The grid lines on the mp re 3 km prt., nd re frm houses. How fr is it from: to to to? 5 If the dimeter of irle is 20 m, find the shortest distne from hord of length 16 m to the entre of the irle. 6 Find the length of plsti oted wire required to mke this lothes line: 3m 7 The irles illustrted hve rdii of length 5 m nd 7 m respetively. Their entres re 18 m prt. Find the length of the ommon tngent m hopstik just fits inside retngulr ox with se 10 m y 15 m. Find the height of the ox. 9 Find y in the following, giving your nswers orret to 3 signifint figures: y m y m 8m tngent 10 m ( y ) m 3m 10 Mrvin the Mgnifient is ttempting to wlk tightrope ross n intersetion from one uilding to nother s illustrted. Using the dimensions given, find the length of the tightrope. 18 m 13 m

22 #endoxedheding 190 The theorem of Pythgors (hpter 8) hllenge 1 highwy runs in n Est-West diretion joining towns nd, whih re 25 km prt. Town lies diretly north from, t distne of 15 km. stright rod is uilt from to the highwy nd meets the highwy t D, whih is equidistnt from nd. Find the position of D on the highwy. 2 nnel nt t wishes to visit ertie eetle t on the opposite vertex of lok of heese whih is 30 m y 20 m y 20 m. Wht is the shortest distne tht nnel must trvel if her fvourite food is heese she htes eting heese? 30 m 20 m 20 m 3 n irrft hnger is semi-ylindril, with dimeter 40 m nd length 50 m. heliopter ples n inelsti rope ross the top of the hnger nd one end is pinned to orner t. The rope is then pulled tight nd pinned t the opposite orner 50 m. Determine the length of the rope. 40 m 4 The rdius of the smll irle is r nd the rdius of the semi-irle is R. Find the rtio r : R, given tht the semi-irles pss through eh other s entres. Note: When irles touh, their entres nd their point of ontt lie in stright line, tht is, they re olliner. 5 The lrger irle touhes the dimeter of the semi-irle t its entre. The irles touh eh other nd the semiirle. The semi-irle hs rdius 10 m. Find the rdius of the smll irle. 10 m

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