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1 esureent n geoetr UNRRETE GE RF toi 19 irle geoetr 19.1 verview Wh lern this? For thousnds of ers huns hve een fsinted irles. ine the first looked upwrds towrds the sun nd oon, whih, fro distne t lest, looked irulr, huns hve reted irulr onuents to nture. The ost fous irulr invention, one tht hs een redited s the ost iportnt invention of ll, is the wheel. holrs s erl s ortes nd lto hve een fsinted with the sheer eut of the properties of irles, nd n sholrs de life s work out of studing the. Eulid ws prol the ost fous of these. It is in irle geoetr tht the onepts of ongruene nd siilrit, studied erlier, hve powerful ontet. Wht do ou know? 1 think List wht ou know out irle geoetr. Use thinking tool suh s onept p to show our list. 2 Ir hre wht ou know with prtner nd then with sll group. 3 shre s lss, rete thinking tool suh s lrge onept p tht shows our lss s knowledge of irle geoetr. Lerning sequene verview ngles in irle Interseting hords, sents nd tngents li qudrilterls Tngents, sents nd hords Review NLINE NLY 19irleGeoetr.indd /08/14 10:04 M

2 UNRRETE GE RF WtH this VIeo The stor of thetis: serhlight I: eles-2022

3 esureent n geoetr UNRRETE GE RF int ngles in irle Introdution irle is set of points tht lie fied distne (the rdius) fro fied point (the entre). In irle geoetr, there re n theores tht n e used to solve proles. It is iportnt tht we re lso le to prove these theores. To prove theore: 1. stte the i of the proof 2. use given infortion nd previousl estlished theores to estlish the result 3. give reson for eh step of the proof 4. stte ler onlusion. rts of irle rt (ne) esription igr entre The iddle point, equidistnt fro ll points on the iruferene. It is usull shown dot nd lelled. iruferene Rdius ieter hord egent The outside length or the oundr foring the irle. It is the irle s perieter. stright line fro the entre to n point on the iruferene stright line fro one point on the iruferene to nother, pssing through the entre stright line fro one point on the iruferene to nother The re of the irle etween hord nd the iruferene. The sller segent is lled the inor segent nd the lrger segent is the jor segent. 806 Mths Quest

4 esureent N geoetr rt (ne) UNRRETE GE RF etor esription igr n re of irle enlosed 2 rdii nd the iruferene r portion of the iruferene Tngent ent stright line tht touhes the iruferene t one point onl hord etended eond the iruferene on one side ngles in irle In the digr t right, hords nd for the ngle. r hs sutended ngle. Theore 1 ode The ngle sutended t the entre of irle is twie the ngle sutended t the iruferene, stnding on the se r. roof: Let R = nd QR = R = = Q (rdii of the se irle re equl) R = nd RQ = M = 2 (eterior ngle of tringle) nd QM = 2 (eterior ngle of tringle) Q = = 2( + ) whih is twie the size of RQ = +. R Q R M Q Topi 19 irle geoetr 19irleGeoetr.indd /08/14 10:04 M

5 esureent N geoetr The ngle sutended t the entre of irle is twie the ngle sutended t the iruferene, stnding on the se r. UNRRETE GE RF Theore 2 ode R ll ngles tht hve their verte on the iruferene nd re sutended the se r re equl. roof: Q Join nd Q to, the entre of the irle. Q Let Q = Q = 2 (ngle t the entre is twie) the ngle t the iruferene) RQ = (ngle t the iruferene is hlf the ngle of the entre) Q = RQ. ngles t the iruferene sutended the se r re equl. The pplition of the first two irle geoetr theores n e seen in the following worked eple. WRKE EXMLE 1 Find the vlues of the pronuerls in the digr t right, giving resons for our nswers. THINK 46 WRITE 1 ngles nd 46 re ngles sutended the se r nd oth hve their verte on the iruferene. = 46 2 ngles nd 46 stnd on the se r. The 46 ngle hs its verte on the iruferene nd hs its verte t the entre. The ngle t the entre is twie the ngle t the iruferene. = 2 46 = 92 Theore 3 ode ngles sutended the dieter, tht is, ngles in seiirle, re right ngles. In the digr t right, Q is the dieter. ngles, nd re right ngles. This theore is in ft speil se of Q Theore 1. roof: Q = 180 (stright line) Let refer to the ngle t the iruferene sutended the dieter. In the figure, ould e t the points where, nd re represented on the digr. Q = 90 (ngle t the iruferene is hlf the ngle t the entre) ngles sutended dieter re right ngles. onstruting tngent There re nuer of ws to onstrut tngent to irle, s eplined using the following steps. 808 Mths Quest irleGeoetr.indd /08/14 10:04 M

6 esureent N geoetr UNRRETE GE RF 1. rw irle of rdius 5 nd entre. 2. rw rdius. 3. ll the point of intersetion of the rdius nd the iruferene,. 4. Etend this rdius through to the point Q, 5 outside the irle. 5. Using nd Q s entres, drw interseting rs ove nd elow the line Q. 6. rw stright line joining the points of intersetion. This line is the tngent. 7. Wht do ou notie out the ngle etween Q nd the tngent? 8. Investigte nother tehnique for onstruting tngent to irle. 9. Write set of instrutions for this ethod of onstruting tngent. Theore 4 ode If rdius is drwn to n point on the iruferene nd tngent is drwn t the se point, then the rdius will e perpendiulr to the tngent. In the digr t right, the rdius is drwn to point,, on the iruferene. The tngent to the irle is lso drwn t. The rdius nd the tngent eet t right ngles, tht is, the ngle t equls 90. WRKE EXMLE 2 Find the vlues of the pronuerls in the digr t right, giving reson for our nswer. THINK 1 ngle z is sutended the dieter. Use n pproprite theore to stte the vlue of z. 2 ngle s is fored tngent nd rdius, drwn to the point of ontt. ppl the orresponding theore to find the vlue of s. Q WRITE z = 90 s = 90 Theore 5 ode The ngle fored two tngents eeting t n eternl point is iseted stright line joining the entre of the irle to tht eternl point. roof: R z s T Topi 19 irle geoetr 809

7 esureent N geoetr UNRRETE GE RF onsider ΔR nd ΔT. R = T (rdii of the se irle re equl) is oon. R = T = 90 (ngle etween tngent nd rdii is 90 ) ΔR ΔT (RH) o R = T nd R = T ( orresponding ngles in ongruent tringles re equl). The ngle fored two tngents eeting t n eternl point is iseted stright line joining the entre of the irle to the eternl point. WRKE EXMLE 3 Given tht nd re tngents to the irle, find the vlues of the pronuerls in the digr t right. Give resons for our nswers. THINK 1 ngles r nd s re ngles fored the tngent nd the rdius, drwn to the se point on the irle. tte their size. 2 In the tringle, two ngles re lred known nd so ngle t n e found using our knowledge of the su of the ngles in tringle. 3 is fored the two tngents, so the line, joining the verte with the entre of the irle, isets this ngle. This ens tht ngles t nd u re equl. 4 Δ nd Δ re siilr tringles. WRITE s = r = 90 Δ: t = 180 t = 180 t = 22 = = t = 22, = u u = 22 In Δ nd Δ r + t + 68 = 180 s + u + q = 180 r = s = 90 (proved previousl) t = u = 22 (proved previousl) q = 68 r 68 q s t u 810 Mths Quest

8 MEUREMENT N GEMETRY UNRRETE GE RF Eerise 19.2 ngles in irle INIVIUL THWY RTIE Questions: 1 e, 2, 3 d, 4 7 NLITE Questions: 1d i, 2 f, 3 f, 4 8, 10, 11 MTER Questions: 1 12 FLUENY 1 WE1 Find the vlues of the pronuerls in eh of the following, giving resons for our nswers. 30 Q 32 d g e h f i R WE2 Find the vlues of the pronuerls in eh of the following figures, giving resons for our nswers. d r s 38 Individul pthw intertivit int-4659 e u t f n 75 UNERTNING 3 WE3 Given tht nd re tngents, find the vlue of the pronuerls in eh of the following, giving resons for our nswers. r 70 w 40 t REFLETIN Wht re the oon steps in proving theore? do do-5391 do do do z s Topi 19 irle geoetr 811

9 esureent N geoetr d UNRRETE GE RF 20 s 70 z rz e f z z Note: There e ore thn one orret nswer. In the digr t right, whih ngle is sutended the se r s? 5 M Note: There e ore thn one orret nswer. Referring to the digr t right, whih of the stteents is true? 2 = = 2 F = = 4 M F RENING Vlues re suggested for the pronuerls in the digr rs t right. is tngent to irle nd is the entre. In 25 eh se give resons to justif suggested vlues. t s = t = 45 r = 45 u u = 65 d = 25 e n = 45 7 et out elow is the proof of this result: The ngle t the entre of irle is twie the ngle t the iruferene stnding on the se r. 6 n F R 812 Q Mths Quest irleGeoetr.indd /08/14 10:04 M

10 esureent n geoetr UNRRETE GE RF op nd oplete the following to show tht Q = 2 RQ. onstrut dieter through R. Let the opposite end of the dieter e. Let R = nd RQ =. R = ( ) R R = ( ) = 2 (eterior ngle equls ) R = Q ( ) QR = ( ) Q = ( ) Now RQ = nd Q =. Therefore Q = 2 RQ. 8 rove tht the segents fored drwing tngents fro n eternl point to irle re equl in length. 9 Use the figure drwn elow to prove tht ngles sutended the se r re equl. R role solving 10 Use our knowledge of tpes of tringles, ngles in tringles nd the ft tht the rdius of irle eets the tngent to the irle t right ngles to prove the following theore: The ngle fored etween two tngents eeting t n eternl point is iseted line fro the entre of the irle to the eternl point. 11 W Y X Z K L Q M Q WX is the dieter of irle with entre t. Y is point on the irle nd WY is etended to Z so tht Y = YZ. rove tht ngle ZX is three ties ngle YZ. do-5395 Topi 19 irle geoetr 813

11 esureent N geoetr UNRRETE GE RF 19.3 Interseting hords, sents nd tngents Interseting hords In the digr elow, hords Q nd R interset t X. Theore 6 ode If the two hords interset inside irle, then the point of intersetion divides eh hord into two segents so tht the produt of the lengths of the segents for oth hords is the se. X QX = RX X or = d R roof: Join R nd Q. onsider ΔRX nd ΔQX. XR = XQ (vertill opposite ngles re equl) RQ = RQ (ngles t the iruferene stnding on the se r re equl) R = Q 1ngles t the iruferene stnding on the se r re equl2 ΔRX ~ ΔQX (equingulr) X X = RX (rtio of sides in siilr tringles is equl) QX or X QX = RX X WRKE EXMLE 4 Find the vlue of the pronuerl. THINK X Q X 1 hords nd interset t X. oint X divides eh hord into two prts so tht the produts of the lengths of these prts re equl. Write this s thetil stteent. WRITE X X = X X 2 Identif the lengths of the line segents. X = 4, X =, X = 6, X = 5 3 ustitute the given lengths into the forul nd solve for. 4 = 6 5 = 30 4 = 7.5 R X d Q 814 Mths Quest

12 esureent N geoetr UNRRETE GE RF Interseting sents In the digr elow, hords nd re etended to for sents X nd X respetivel. The interset t X. X Theore 7 ode If two sents interset outside the irle s shown, then the following reltionship is lws true: X d X X = X X or = d. roof: Join nd to, the entre of the irle. Let =. = 2 (ngle t the entre is twie the ngle t the iruferene stnding on the X se r) Refle = (ngles in revolution dd to 360 ) = 180 (ngle t the entre is twie the ngle t the iruferene stnding on the se r) X = (ngle su of stright line is 180 ) = X onsider ΔX nd ΔX. X is oon. = X 1 shown previousl 2 X = X 1ngle su of tringle is ΔX ~ ΔX 1 equingulr 2 X X = X X or X X = X X WRKE EXMLE 5 Find the vlue of the pronuerl. X Topi 19 irle geoetr 815

13 esureent N geoetr UNRRETE GE RF THINK 1 ents X nd X interset outside the irle t X. Write the rule onneting the lengths of X, X, X nd X. 2 tte the length of the required line segents. 3 ustitute the length of the line segents nd solve the eqution for. WRITE X X = X X X = + 6 X = 6 X = X = 7 = 12 ( + 6) 6 = = 84 6 = 48 = 8 Interseting tngents In the following digr, tngents nd interset t nd =. Theore 8 ode If two tngents eet outside irle, then the lengths fro the eternl point to where the eet the irle re equl. roof: Join nd to, the entre of the irle. onsider Δ nd Δ. is oon. = (rdii of the se irle re equl) = (rdius is perpendiulr to tngent through the point of ontt) Δ Δ (RH) = (orresponding sides of ongruent tringles re equl). If two tngents eet outside irle, the lengths fro the eternl point to the point of ontt re equl. WRKE EXMLE 6 Find the vlue of the pronuerl. THINK 3 WRITE 1 nd re tngents interseting t. = tte the rule tht onnets the lengths nd. 2 tte the lengths of nd. =, = 3 3 ustitute the required lengths into the eqution to find the vlue of. = Mths Quest

14 esureent N geoetr hords nd rdii In the digr elow, the hord nd the rdius interset t X t 90 ; tht is, X = 90. isets the hord ; tht is, X = X. UNRRETE GE RF Theore 9 ode If rdius nd hord interset t right ngles, then the rdius isets the hord. roof: Join nd. onsider ΔX nd ΔX. = (rdii of the se irle re equl) X = X (given) X is oon. ΔX ΔX (RH) X = X (orresponding sides in ongruent tringles re equl) If rdius nd hord interset t right ngles, then the rdius isets the hord. The onverse is lso true: X If rdius isets hord, the rdius nd the hord eet t right ngles. Theore 10 hords equl in length re equidistnt fro the entre. This theore sttes tht if the hords MN nd R re of equl length, then =. roof: onstrut MN nd R. Then isets MN nd isets R (Theore 9) euse MN = R, M = N = = R. onstrut M nd, nd onsider ΔM nd Δ. M = (shown ove) M = (rdii of the se irle re equl) M = = 90 ( onstrution) ΔM Δ (RH) o = (orresponding sides in ongruent tringles re equl) hords equl in length re equidistnt fro the entre. WRKE EXMLE 7 Find the vlues of the pronuerls, given tht =. X G E 3 n 2.5 F H M N M N R R Topi 19 irle geoetr 817

15 esureent n geoetr UNRRETE GE RF think 1 ine the rdius G is perpendiulr to the hord, the rdius isets the hord. WrIte E = E 2 tte the lengths of E nd E. E =, E = 3 3 ustitute the lengths into the eqution to find the = 3 vlue of. 4 nd re hords of equl length nd E nd E = F F re perpendiulr to these hords. This iplies tht E nd F re equl in length. 5 tte the lengths of E nd F. E = n, F = ustitute the lengths into the eqution to find the vlue of n. n = 2.5 The iruentre of tringle In the digr, irle psses through the verties of the tringle. The irle is lled the iruirle of tringle, nd the entre of the irle is lled the iruentre. The iruentre is loted s follows. rw n tringle. Lel the verties. onstrut perpendiulr isetors of, nd, nd let the isetors interset t. This ens tht = =, so irle n e drwn through, nd with entre t. Eerise 19.3 Interseting hords, sents nd tngents InIVIuL thws refletion Wht tehniques will ou use to prove irle theores? rtise Questions: 1 7, 10 onsolite Questions: 1 8, 10 ster Questions: 1 11 Individul pthw intertivit int Mths Quest

16 esureent N geoetr FLUENY 1 UNRRETE GE RF Find the vlue of the pronuerl in eh of the following. WE4 4 6 X X 4 X 2 WE5 9 Find the vlue of the pronuerl in eh of the following n 5 d n 7 3 WE6 Find the vlue of the pronuerls in eh of the following WE7 Find the vlue of the pronuerl in eh of the following d UNERTNING 5 Note: There e ore thn one orret nswer. In whih of the following figures is it possile to find the vlue of through solving liner eqution? M Topi 19 irle geoetr 19irleGeoetr.indd /08/14 10:05 M

17 esureent N geoetr UNRRETE GE RF 6 Find the length, T, in the digr elow. RENING Q rove the result: If rdius isets hord, then the rdius eets the hord t right ngles. Reeer to provide resons for our stteents. 8 rove the result: hords tht re n equl distne fro the entre re equl in length. rovide resons for our stteents. 9 rove tht the line joining the entres of two interseting irles isets their oon hord t right ngles. rovide resons for our stteents. RLEM LVING 10 lulte the pronuerl for eh of the following digrs is the dieter of the irle. is hord perpendiulr to nd eeting t M. R T 4 Wh is M the idpoint of? If M =, M = nd M =, prove tht 2 =. Eplin wh the rdius of the irle is equl to M Mths Quest

18 esureent n geoetr H HLLenge 19.1 UNRRETE GE RF 19.4 li qudrilterls Qudrilterls in irles li qudrilterl hs ll four verties on the iruferene of irle; tht is, the qudrilterl is insried in the irle. In the digr t right, points,, nd lie on the iruferene; hene, is li qudrilterl. It n lso e sid tht points,, nd re onli; tht is, the irle psses through ll the points. Theore 11 ode The opposite ngles of li qudrilterl re suppleentr (dd to 180 ). roof: Join nd to, the entre of the irle. Let =. Refle = 2 ( ngle t the entre is twie the ngle t the iruferene stnding on the se r) Refle = (ngles in revolution dd to 360 ) = 180 ( ngle t the entre is twie the ngle t the iruferene stnding on the se r) + = 180 iilrl, + = 180. pposite ngles in li qudrilterl re suppleentr. The onverse is lso true: If opposite ngles of qudrilterl re suppleentr, then the qudrilterl is li. WorKe exle 8 Find the vlues of the pronuerls in the digr elow. Give resons for our nswers. 120 Q 75 R Topi 19 irle geoetr 821

19 esureent n geoetr UNRRETE GE RF think 1 QR is li qudrilterl, so its opposite ngles re suppleentr. First find the vlue of onsidering pir of opposite ngles QR nd R nd foring n eqution to solve. 2 Find the vlue of onsidering the other pir of opposite ngles ( Q nd QR). WrIte QR + R = 180 (The opposite ngles of li qudrilterl re suppleentr.) QR = 75, R = + 75 = 180 = 105 Q + QR = 180 Q = 120, QR = = 180 = 60 Theore 12 ode The eterior ngle of li qudrilterl is equl to the interior opposite ngle. T roof: Q + QR = 180 (opposite ngles of li qudrilterl) Q + T = 180 (djent ngles on stright line) Therefore T = QR. The eterior ngle of li qudrilterl is equl to the interior opposite ngle. WorKe exle 9 Find the vlue of the pronuerls in the digr elow. think 50 1 is li qudrilterl. The eterior ngle,, is equl to its interior opposite ngle,. 2 The eterior ngle, 100, is equl to its interior opposite ngle,. 100 WrIte Eerise 19.4 li qudrilterls InIVIuL thws =, = 50 o = 50. = 100, = o = 100. Q R refletion Wht is li qudrilterl? rtise Questions: 1 6, 8 onsolite Questions: 1 8 ster Questions: 1 9 Individul pthw intertivit int Mths Quest

20 esureent n geoetr FLuen 1 WE8 Find the vlues of the pronuerls in eh of the following. UNRRETE GE RF d e 2 WE9 Find the vlues of the pronuerls in eh of the following. d e M Note: There e ore thn one orret nswer. Whih of the following orretl sttes the reltionship etween, nd z in the digr shown? = nd = 2z z = 2 nd = 2z f f n n = 2 nd + z = = 180 nd z = 2 unerstning 4 The steps elow show ou how to set out the proof tht the opposite ngles of li qudrilterl re equl. Find the size of. Find the size of the refle ngle. Find the size of. d Find +. 5 M Note: There e ore thn one orret nswer. Whih of the following stteents is lws true for the digr shown? r = t r = p r = q r = s Whih of the following stteents is orret for the digr shown? r + p = 180 q + s = 180 t + p = 180 t = r t q p z r s do-5396 Topi 19 irle geoetr 823

21 esureent n geoetr UNRRETE GE RF do-5397 resoning 6 rove tht the eterior ngle of li qudrilterl is equl to the interior opposite ngle. 7 lulte the vlues of the pronuerls in eh of these digrs. z role solving 8 lulte the vlue of eh pronuerl in the digr t right. 9 F 70 E F = 70, EF =, E = nd =. Find the vlues of, nd. rove tht is prllel to F z + 5 z w 824 Mths Quest

22 esureent N geoetr UNRRETE GE RF 19.5 Tngents, sents nd hords The lternte segent theore onsider the figure shown. Line is tngent to the irle t the point. line is drwn fro to nwhere on the iruferene, point. The ngle defines segent (the shded re). The unshded prt of the irle is lled the lternte segent to. Now onsider ngles sutended the hord in the lternte segent, suh s the ngles rked in red nd lue. The lternte segent theore sttes tht these re equl to the ngle tht de the segent, nel: = E nd = F Theore 13 ode The ngle etween tngent nd hord is equl to the ngle in the lternte segent. roof: We re required to prove tht = F. onstrut the dieter fro through, eeting the irle t G. Join G to the points nd F. G = G = 90 (rdii tngent t point of ontt) GF = 90 (ngle in seiirle is 90 ) G = 90 (ngle in seiirle is 90 ) onsider ΔG. We know tht G = 90. G + G + G = G + G = 180 G + G = 90 G is lso right ngle. Equte the two results. nel the equl ngles on oth sides. G = + G = 90 G + G = + G G = Now onsider the ft tht oth tringles G nd F re sutended fro the se hord (). G = F (ngles in the se segent stnding on the se r re equl. Equte the two equtions. F = E G F F Topi 19 irle geoetr 825

23 esureent N geoetr UNRRETE GE RF WRKE EXMLE 10 Find the vlue of nd, giving resons. THINK 62 T WRITE 1 Use the lternte segent theore to find. = 62 (ngle etween tngent nd hord is equl to the ngle in the lternte segent) 2 The vlue of is the se s euse nd re sutended the se hord T. Tngents nd sents Theore 14 ode If tngent nd sent interset s shown, the following reltionship is lws true: X X = (XT) 2 or = 2. roof: Join T nd T. onsider ΔTX nd ΔXT. TX is oon. XT = XT (ngle etween tngent nd hord is equl to the ngle in the lternte segent) XT = XT (ngle su of tringle is 180 ) ΔTX~ΔXT (equingulr) o X XT = XT X or X X = (XT) 2. WRKE EXMLE 11 Find the vlue of the pronuerl. THINK 1 ent X nd tngent XT interset t X. Write the rule onneting the lengths of X, X nd XT. T 8 5 = 62 (ngles in the se segent stnding on the se r re equl) X WRITE X X = (XT) 2 X X T T 826 Mths Quest

24 esureent n geoetr UNRRETE GE RF 2 tte the vlues of X, X nd XT. X = + 5, X = 5, XT = 8 3 ustitute the vlues of X, X nd XT into the eqution nd solve for. ( + 5) 5 = = 64 5 = 39 = 7.8 Eerise 19.5 Tngents, sents nd hords InIVIuL thws rtise Questions: 1, 2, 4 6, 8, 10, 13 15, 17, 20, 22 onsolite Questions: 1 3, 5, 7, 9, 11, 13, 14, 16, 19, 21, 22 FLuen 1 WE10 Find the vlue of the pronuerls in the following WE11 Find the vlue of the pronuerls in the following. 5 p 4 Individul pthw intertivit int Line is tngent to the irle s shown in the figure on the right. Find the vlues of the ngles lelled nd. Questions 4 to 6 refer to the figure on the right. The line MN is tngent to the irle, nd E is stright line. The irles hve the se rdius ster Questions: Find 6 different right ngles. 5 M If = 20, then F nd FG re respetivel: 70 nd nd nd nd 70 6 M tringle siilr to F is: FG FG E GE q E F 21 G M N refletion esrie the lternte segent of irle. Topi 19 irle geoetr 827

25 esureent N geoetr 7 Find the vlues of the ngles nd in the figure t right. UNRRETE GE RF UNERTNING 8 how tht if the su of the two given ngles in question 7 is 90, then the line ust e dieter. 9 Find the vlue of in the figure t right, given tht the line underneth the irle is tngent. 10 In the figure t right, epress in ters of nd. This is the se drwing s in question Two tngent lines to irle eet t n ngle, s shown in the figure t right. Find the vlues of the ngles, nd z. 12 olve question 11 in the generl se (see the figure t right) nd show tht = 2. This result is iportnt for spe nvigtion (igine the irle to e the Erth) in tht n ojet t n e seen people t nd z t the se tie. 13 In the figure t right, find the vlues of the ngles, nd z. 14 M Eine the figure t right. The ngles nd (in degrees) re respetivel: 51 nd nd nd nd z z z Mths Quest

26 esureent N geoetr UNRRETE GE RF Questions 15 to 17 refer to the figure t right. The line is tngent to the irle t point. Line is hord tht eets the tngent t. 15 Find the vlues of the ngles nd. 16 M The tringle whih is siilr to tringle is: 17 M The vlue of the ngle z is: RENING 18 Find the vlues of the ngles, nd z in the figure t right. The line is tngent to the irle t. 19 Find the vlues of the ngles, nd z in the figure t right. The line is tngent to the irle t. The line is dieter. 20 olve question 19 in the generl se; tht is, epress ngles, nd z in ters of (see the figure t right). 21 rove tht, when two irles touh, their entres nd the point of ontt re olliner. RLEM LVING 22 Find the vlue of the pronuerls in the following. 5 4 n 4 6 d k z z z z Topi 19 irle geoetr 829

27 MEUREMENT N GEMETRY e 1 f UNRRETE GE RF Find the vlues of, nd in eh se. E = 50 nd E = E 50 E H HLLENGE w 70 F Mths Quest

28 esureent n geoetr UNRRETE GE RF NLINE NLY 19.6 Review The Mths Quest Review is ville in ustoisle fort for students to deonstrte their knowledge of this topi. The Review ontins: Fluen questions llowing students to deonstrte the skills the hve developed to effiientl nswer questions using the ost pproprite ethods role solving questions llowing students to deonstrte their ilit to ke srt hoies, to odel nd investigte proles, nd to ounite solutions effetivel. sur of the ke points overed nd onept p sur of this topi re ville s digitl douents. Review questions ownlod the Review questions douent fro the links found in our eooklu. Lnguge int-2880 int-2881 lternte segent theore ngle r hord irle iruentre iruirle iruferene onli li li qudrilterl dieter jor segent inor segent rdius sent setor segent sutend tngent theore int-3894 Link to ssessn for questions to test our rediness For lerning, our progress s ou lern nd our levels of hieveent. ssessn provides sets of questions for ever topi in our ourse, s well s giving instnt feedk nd worked solutions to help iprove our thetil skills. the stor of thetis is n elusive Jrnd video series tht eplores the histor of thetis nd how it helped shpe the world we live in tod. <Tet to oe> Topi 19 irle geoetr 19irleGeoetr.indd /08/14 10:06 M

29 <InVestIgtIon> For rih tsk or <esureent n geoetr> For uzzle InVestIgtIon UNRRETE GE RF rih tsk Vrition of distne 60 N 45 N 30 N 15 ut 90 E or 105 E Eq 75 E N W E E 135 E 120 E N N 60 N N Eq ut 15 or N E E E W E 135 E E E 90 E Ltitude 30 N Longitude North ole 0 100E r 0 Equto 2 1 outh ole The distne (in k) etween two points on the se line of ltitude is given the forul: istne = ngle setor etween the two points 111 os 1 degree of ltitude Mths Quest irleGeoetr.indd /08/14 10:06 M

30 esureent n geoetr UNRRETE GE RF 1 The size of the ngle setor etween 1 nd 2 is 100 nd these two points lie on 0 ltitude. The distne etween the points would e lulted s os 0. eterine this distne. 2 Move the two points to the 10 line of ltitude. lulte the distne etween 1 nd 2 in this position. Round our nswer to the nerest kiloetre. 3 oplete the following tle showing the distne (rounded to the nerest kiloetre) etween the points 1 nd 2 s the ove fro the equtor towrds the pole. 4 esrie wht hppens to the distne etween 1 nd 2 s we ove fro the equtor to the pole. Is there onstnt hnge? Eplin. 5 You would perhps ssue tht, t ltitude of 45, the distne etween 1 nd 2 is hlf the distne etween the points t the equtor. This is not the se. t wht ltitude does this our? 6 n the grid lines provided, sketh grph displing the hnge in distne etween the points in oving fro the equtor to the pole. 7 onsider the points 1 nd 2 on lines of longitude seprted 1. n wht line of ltitude (to the nerest degree) would the points e 100 k prt? 8 Keeping the points 1 nd 2 on the se line of ltitude, nd vring their lines of longitude, investigte the rte tht the distne etween the hnges fro the equtor to the pole. Is it ore or less rpid in oprison to wht ou found erlier? Topi 19 irle geoetr 833

31 <InVestIgtIon> esureent n For geoetr rih tsk or <esureent n geoetr> For uzzle oe uzzle UNRRETE GE RF Where is 2 litres of hdrohlori id produed eh d? 13 0º The vlues of the lettered ngles give the puzzle s nswer ode. 80º 85º M N 75º 0º 11 I Y 92 R 107º º 69º 81º 7º 115º 0º 6 11 E T G H 100º 125º 105º 115º 834 U 88º 95º 111º 110º 105º 88º 99º 50º 99º 73º 105º 125º 120º 115º 117º 88º 125º 80º 115º 110º 105º 90º 88º 100º 95º Mths Quest irleGeoetr.indd /08/14 10:07 M

32 esureent n geoetr tivities UNRRETE GE RF 19.1 overview Video The stor of thetis (eles-2022) 19.2 ngles in irle Intertivities ngles in irle (int-2795) I intertivit 19.2 (int-4659): ngles in irle igitl dos killheet (do-5390): Using tests to prove ongruent tringles killheet (do-5391): orresponding sides nd ngles of ongruent tringles killheet (do-5392): Using tests to prove siilr tringles killheet (do-5393): ngles in tringle killheet (do-5394): More ngle reltions WorkHEET 19.1 (do-5395): irle geoetr I 19.3 Interseting hords, sents nd tngents Intertivit I intertivit 19.3 (int-4660): Interseting hords, sents nd tngents to ess eooklus tivities, log on to 19.4 li qudrilterls Intertivit I intertivit 19.4 (int-4661): li qudrilterls igitl dos killheet (do-5396): ngles in qudrilterl WorkHEET 19.2 (do-5397): irle geoetr II 19.6 review Intertivities Word serh (int-2880) rossword (int-2881) udoku (int-3894) I intertivit 19.5 (int-4662): Tngents, sents nd hords igitl dos hpter sur (do-13821) onept p (do-13822) Topi 19 irle geoetr 835

33 esureent N geoetr nswers topi 19 irle geoetr UNRRETE GE RF Eerise 19.2 ngles in irle 1 = 30 (theore 2) = 25, = 25 (theore 2 for oth ngles) = 32 (theore 2) d = 40, = 40 (theore 2 for oth ngles) e = 60 (theore 1) f = 40 (theore 1) g = 84 (theore 1) h = 50 (theore 2); = 100 (theore 1) i = 56 (theore 1) 2 s = 90, r = 90 (theore 3 for oth ngles) u = 90 (theore 4); t = 90 (theore 3) = 90, n = 90 (theore 3 for oth ngles) d = 52 (theore 3 nd ngle su in tringle = 180 ) e = 90 (theore 4) f = 90 (theore 4); = 15 (ngle su in tringle = 180 ) 3 = z = 90 (theore 4); = w = 20 (theore 5 nd ngle su in tringle = 180 ) s = r = 90 (theore 4); t = 140 (ngle su in qudrilterl = 360 ) = 20 (theore 5); = z = 70 (theore 4 nd ngle su in tringle = 180 ) d s = = 90 (theore 4); = 70 (theore 5); r = z = 20 (ngle su in tringle = 180 ) e = 70 (theore 4 nd ngle su in tringle = 180 ); = z = 20 (ngle su in tringle = 180 ) f = = 75 (theore 4 nd ngle su in tringle = 180 ); z = 75 (theore 1) 4 5, 6 se ngles of right-ngled isoseles tringle r + s = 90, s = 45 r = 45 u is the third ngle in Δ, whih is right-ngled. d is the third ngle in Δ, whih is right-ngled. e nd F stnd on the se r with t the entre nd F t the iruferene. 7 R = (rdii of the irle) R = (equl ngles lie opposite equl sides) = 2 (eterior ngle equls the su of the two interior opposite ngles) R = Q (rdii of the irle) QR = (equl ngles lie opposite equl sides) Q = 2 (eterior ngle equls the su of the two interior opposite ngles) Now RQ = + nd Q = = 2( + ). Therefore Q = 2 RQ. 8, 9 hek with our teher. 10 hek with our teher. 11 hek with our teher. Eerise 19.3 Interseting hords, sents nd tngents 1 = 3 = 3 = 6 2 n = 1 = 7.6 n = 13 d = 4 3 = 5 = 7 = 2.5, = = 2.8 = 3.3 = 5.6 d = 90 5,, 6 T = 3 7, 8, 9 hek with our teher. 10 = 3"2 = 6 = 3, = hek with our teher. hllenge 19.1 Eerise 19.4 li qudrilterls 1 = 115, = 88 = 85 n = 25 d = 130 e = = 90 f = 45, = 95 2 = 85, = 80 = 110, = 115 = 85 d = 150 e = 90, = 120 f = 120, n = d 180 5,,, 6 hek with our teher. 7 = 93, = 87, z = 93 = 2 or 5 2 = 2 3 or w = 110, = 70, = 140, z = = 110, = 70 nd = 110 hek with our teher. Eerise 19.5 Tngents, sents nd hords 1 = 70 = 47, = 59 2 p = 6 q = 8 3 = 42, = M, N, F, F, EG, EG = 42, = 62 8 nswers will vr = = 80, = 20, z = nswers will vr. 13 = 85, = 20, z = = 50, = = 33, = 55, z = = 25, = 65, z = =, = 90, z = hek with our teher. 22 = 5 k = 12 = 6, n = 6 d = 7 e = 4, = 2 f w = 3, = 5 23 = 50, = 50 nd = 80 = 50, = 70 nd = Mths Quest

34 esureent N geoetr UNRRETE GE RF hllenge Investigtion Rih tsk k k Ltitude istne etween 1 nd 2 (k) The distne etween 1 nd 2 dereses fro k t the equtor to 0 k t the pole. The hnge is not onstnt. The distne etween the points dereses ore rpidl on oving towrds the pole. 5 Ltitude istne etween 1 nd 2 (k) Ltitude 7 Ltitude 26 8 nswers will vr. Teher to hek. ode puzzle In our stoh to id digestion Topi 19 irle geoetr 837