Lesson-5 PROPERTIES AND SOLUTIONS OF TRIANGLES
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1 Leon-5 PROPERTIES ND SOLUTIONS OF TRINGLES Reltion etween the ide nd trigonometri rtio of the ngle of tringle In ny tringle, the ide, oppoite to the ngle, i denoted y ; the ide nd, oppoite to the ngle nd repetively, re denoted y nd. Formule involving ide nd ngle of tringle. Sine Rule : In ny tringle, R in in in where R i the rdiu of the irumirle of the tringle. R i known irumrdiu of the tringle.. oine Rule : In ny tringle, o o o or o + 0 or o + 0 or o + 0. Hlf -ngle Formule in in in where i the emiperimeter of tringle i.e. o, o, o tn, tn, tn
2 . Projetion Rule : In ny tringle, o + o o + o o + o 5. Npier nlogy : In ny tringle, tn ot tn ot tn ot 6. re of tringle : The re of tringle i given y in in in R r Hero formul where r rdiu of inirle of tringle inrdiu of tringle irumirle, Inirle nd Ex-irle of Tringle The irle whih pe through the ngulr point vertie of tringle i lled it irumirle. It rdiu i denoted y R R in in in O lo, R
3 Inirle : The irle whih n e inried within the tringle o to touh eh of the ide i lled it inried irle or inirle. It rdiu i denoted y r. r i lled inrdiu of tringle. r tn tn tn F E lo, r R in in in r D Eried irle : The irle whih touhe the ide nd the two ide nd produed of tringle i lled it eried irle, oppoite the ngle. It rdiu i denoted y r. Similrly r denote the rdiu of the irle whih touhe the ide nd the two ide nd produed. lo r denote the rdiu of the irle touhing ide nd the two ide nd produed. r tn R in o o r tn R o in o r r tn R o o in l r + r + r r + R r r + r r + r r S r r r rs r R Equlity hold for only n equilterl tringle. in in in 8. Equlity hold for only n equilterl tringle. Ptolemy Theorem If D i yli qudrilterl, then.d.d +.D
4 SOLVED EXMPLES Ex.: Prove tht o in. Sol.: in in in, [uing ine Rule] in o in o o o in o, [ + + ] o in + in o. Ex.: In ny tringle prove tht, ot + ot + ot 0. Sol.: in R in in ot + ot + ot R o o o R.. [ + + R ]. 0
5 Ex.: In tringle, prove tht, ot + ot + ot. Sol.: L.H.S. ot + ot + ot R.H.S. Ex.: In ny tringle, prove tht, + o + + o + + o + +. Sol.: L.H.S. + o + + o + + o o + o + o + o + o + o, o + o + o + o + o + o + + R.H.S. [y uing projetion Rule] Ex.5: If p, p, p re the ltitude of the tringle, prove tht, R p p p o o o. Sol.: In the tringle, let N p M p Q p Q M N Then, re of tringle p p p or p, p, p o o o o o o p p p, o in o in o in R R R [uing ine rule]
6 R in in in R in o in o R in o o [uing + + ] R in in in R in in in R. R R. R [uing ine rule] R R R R [y uing R ] Ex.6: In ny, if, nd 60º, olve the tringle. Sol.: Two ide nd inluded ngle i given tn ot ot 0º tn 60º tn 5º tn 60º tn 5º 5º tn 60º 5º tn 5º or 0º...i We know, º + 0º...ii Solving i nd ii, we get 75º & 5º To find ide, we ue ine Rule in in 60º
7 or 6 Thu 5º, 75º nd 6. Ex.7: If 0º, 00, 00, olve the tringle. Sol.: in in in ; 5 or ; 5 5 ; in in in in Ex.8: Let O e point inide tringle uh tht O O O. Show tht ot ot + ot + ot. Sol.: O nd O pplying the ine Rule in tringle O, we hve in O in - O in in pplying the ine Rule in tringle O, we hve O in in...i - O - O in in...ii From i nd ii, we hve in in in in Uing Sine Rule we hve R in in in in in in R in in in in in in in in in ot ot ot + ot ot ot or ot ot + ot + ot.
8 Ex.9: If the ide of tringle re in.p nd if it gretet ngle exeed the let ngle y, how tht the ide re in the rtio x : : + x where x Sol.: Let the ide e d,, + d : d > 0 Let e the let ngle nd e the gretet ngle. Let ; then + nd 80º o 7 o. d in d in in[ ] d or in d in in From firt nd eond term we get in in...i d in d in d in or d in y ompenendo nd dividendo, we hve + d d d in in in in in o o in or d tn tn tn d tn...ii From third nd fourth term of i we get in in in o o o o Note: 0 < < o > 0 ; i ute tn o o...iii
9 From ii nd iii we get d in o d in o o o o 7 o x Required Rtio d : : + d d/ : : + d/ x : : + x. Ex.0: In tringle, the medin D nd the perpendiulr E from the vertex to the ide divide ngle into three equl prt. Show tht o in. Sol.: D DE E D D DE E D In D o [Sine tringle DE nd tringle E re ongruent] 8...i In tringle, we hve o o o...ii D E
10 Sutrting ii from i we get o o o 8 o o 8 o in. Ex.: Sol.: If, nd re the ditne of the vertie of tringle from neret point of ontt of the inirle with ide of, prove tht r Given. F E F D D E F E Perimeter of tringle I D + + v re of tringle r r Ex.: In ny tringle, if o etween 0 nd, prove tht tn tn tn tn tn tn. where,, lie Sol.: o tn tn
11 y omponendo nd dividendo tn tn Similrly, tn nd tn tn tn tn tn tn tn...i Now tn tn tn From i nd ii...ii tn tn tn tn tn tn. Ex.: The ietor of ngle of tringle meet t D. If D l then, prove tht i l o ii l Sol.: i re of tringle re of tringle D + re of tringle D in l in l in o l + l o ii D D D
12 or D D In tringle D, D o D D l D l... l o + + l D Sutituting vlue of o from i we get D l l + + l D Eqution give l l or or l. Ex.: yli qudrilterl D of re nd i ute nd the digonl D i inried in unit irle. If one of it ide find the length of the other ide. Sol.: Given, D In tringle D, O O OD O R O eing entre of the irle D in R in Given irle i irumirle of D Hene Uing oine Rule in tringle D D i yli qudrilterl o D D. D D D or D D 0
13 or D D + 0 D Uing oine Rule in tringle D, we hve o D D. D D. D or + D + D 0...i re of yli qudrilterl re of tringle D + re of tringle D +.D.. in.. D in or.d...ii Solving i nd ii, we get D Hene length of ide of yli qudrilterl re D, D.
14 OJETIVE SSIGNMENT hooe the orret option in the following :. If,, 5, then the vlue of in i. If 5 o o o 5, then the tringle i d none of thee right ngled ioele otue ngled d equilterl. If,, 60º then i the root of the eqution d o + o + o d 5. In tringle, if, then tn i equl to d 6. If the ide of tringle re : 7 : 8 then R : r : 7 7 : : 7 d 7 : 7. In, R r o o o re of tringle d re of tringle 8. If p, p, p re repetively the perpendiulr from the vertie of tringle to the oppoite ide, then p p p 8 R 8R 8 R d none of thee 9. If, nd, then the numer of tringle tht n e ontruted i Infinite Two One d Nil 0. If one ide of tringle i twie the other ide nd the ngle oppoite to thee ide differ y 60º, then the tringle i Equilterl Ioele Right ngled d none of thee. If in tringle, o o o then the vlue of the ngle i 90º 60º 0º d none of thee
15 . In tringle, the length of the two lrger ide re 0 nd 9 repetively. If the ngle re in.p, then the length of the third ide n e d 5 6. In right-ngled tringle the hypotenue i four time long the ditne of the hypotenue from the oppoite vertex. It ute ngle re 0º, 60º 5º, 75º 5º, 5º d none of thee. irle i inried in n equilterl tringle of ide 6. The re of ny qure inried in the irle i d none of thee 5. If in tringle,, then ot.tn / / d none of thee 6. If in tringle, the exrdii r, r, r re uh tht r r r then : : i 5 : : : 5 : : 5 d none of thee 7. If tringle i right ngled t then the dimeter of the inried irle of the tringle i d none of thee 8. In tringle, tn nd tn /. If 65 then the irumrdiu of the tringle i 65 / 7 65/ 65 d none of thee 9. If the medin D of tringle mke n ngle with, then in i equl to in in in d none of thee 0. If the ietor of ngle of tringle mke n ngle with, then in i equl to o in in. The expreion i equl to d none of thee o in o o o d none of thee. In tringle, point D nd E re tken on ide uh tht D DE E. If DE ED, then tn tn tn tn + 6 tn 6 tn tn d 9 ot tn 0
16 . If the ngle, nd of tringle re in.p. nd the ide,, oppoite thee ngle re in G.P., then, nd re in G.P..P. H.P. d none of thee. The length of the ide of the tringle tify, , then tringle i Right ngled Ioele Equilterl d none of thee 5. If the ngle,, of re in.p., then d none of thee 6. The rtio of the ditne of the orthoentre of n ute-ngled from the ide, nd i o : o : o e : e : e in : in : in d none of thee 7. In, I i the inentre. The rtio I : I : I i equl to o e : o e : oe in : in : in e : e :e d none of thee 8. If,, re the ltitude of nd ; denote it perimeter then + + i equl to. d none of thee 9. In D, the ide re in the rtio : 5 : 6. The rtio of the iumrdiu nd the inrdiu i 8 : 7 : 7 : d 6 : 7 0. In n equilterl tringle, irumrdiu : inrdiu : exrdiu i equl to : : : : : : d : : MORE THN ONE ORRET NSWERS. If in, 6, nd o then 5 in r 9 d none of thee 5. The numer of poile tringle in whih m, m nd 60 i 0 d none of thee
17 . In, nd : :. If tn, 0, then d 60. In tringle the oine of two ngle re inverely proportionl to the ide oppoite the ngle. The tringle i ioele equilterl right ngled d none of thee 5. In D, the line igement D, E nd F re three ltitude. If R i the irumrdiu of the D, ide of the DDEF will e R in o in d o 6. In, tn nd tn re the root of the eqution x + x, where, nd re the ide of the tringle. Then tn ot 0 in + in d none of thee 7. The ditne of the irumentre of the ute-ngled from the ide, nd re in the rtio in : in : in o : o : o ot : ot : ot d none of thee 8. In ny, in i equl to in in o o 0 d none of the 9. In, o. Then,, re in P d + 0. In, tn < 0. Then tn. tn < tn. tn > tn +tn + tn < 0 d tn + tn + tn > 0
18 omprehenion-i MISELLNEOUS SSIGNMENT In tringle, the um of two ide i x nd their produt i y uh tht x + zx z y where z i the third ide of the tringle.. Gretet ngle of the tringle i d 5. Squre of the length of the third ide of the tringle i x + y x y x + y d x y. re of the tringle i y x x y /y 8 /x d x z. Inrdiu of the tringle i x z y y x z x y d z y omprehenion-ii If in tringle with re, r r r ; D i the mid-point of, D. DL i perpendiulr to, re of the tringle LD i. 5. in i equl to d 5 6. DL i equl to d D 7. i equl to d 8 Mth the following 8. In tringle. o + o p R + r. o + o + o q r. ot + ot + ot r R in in in D. R in / in / in /
19 9.. If,, e the length of medin of tringle p then + + i equl to. Let the point P liein the interior of n equilterl q tringle of ide length nd it ditne from the ide, nd re repetively x, y nd z then x + y + z i equl to. In tringle,,, re in P nd,, re in q / GP, then i equl to D. In tringle,, the let vlue of i INTEGER TYPE QUESTIONS 0. In the vlue of R i. In the vlue of. In the vlue of r r r r i r r r r r r i. In the vlue of. In the vlue of 5. In the vlue of r r r r R Rin i o i i 6. In if S repreent the re then in / o / S i 7. In r r R i 8. In if,, 5, then r r r 9. In ny if, 60, then I I i i
20 PREVIOUS YER QUESTIONS IIT-JEE/JEE-DVNE QUESTIONS.,, re the ide of uh tht no two ide re equl nd x x if x i rel then rnge for R i 7, d 7,. In,,, re the length of it ide nd,, re the ngle of tringle. The orret reltion i given y in o o in + in o d o in. mn from the top of 00 metre high tower ee r moving towrd the tower t n ngle of depreion of 0. fter ome time, the ngle of depreion eome 60. The ditne in metre trvelled y the r during thi time i d 00. pole tnd inide tringulr prk. If the ngle of elevtion of the top of the pole from eh orner of the prk i me, then in the foot of the pole i t the entroid irum entre inentre d other entre 5. The minimum vlue of the expreion in + in + in, where,, re rel poitive ngle tifying + + i poitive zero negtive d 6. If the vertie P, Q, R of tringle PQR re rtionl point, whih of the following point of the tringle PQR i re lwy rtionl point? entroid inentre irumentre d orthoentre in D 7. In tringle,, nd D divide internlly in the rtio :, then in D 6 d 8. Whih of the following piee of dt doe not uniquely determine n ute ngle tringle
21 R eing the rdiu of the irumirle?, in, in,,, in, R d, in, R 9. In tringle, D i the ltitude from. Given >, nd D then, d none of thee 0. In tringle, in d. Let 0 5 e regulr hexgon inried in irle of unit rdiu. The produt of the length of the line egment 0, 0 nd 0 i d. In tringle, if, 0, then the tringle i right ngled ioele otue ngled d none of thee. In tringle, the length of the two lrger ide re 0 nd 9 repetively. If the ngle re in.p., then the length of the third ide n e d In tringle if then ngle i equl to d 5 5. The ide of tringle re in the rtio : :, then the ngle of the tringle re in the rtio : : 5 : : : : d : : 6. In n equilterl tringle, oin of rdii unit eh re kept o tht they touh eh other nd lo the ide of the tringle. re of the tringle i d If i ioele tringle nd one of ngle i 0º nd the rdiu of it inirle i of length then
22 the re of i d 7 8. Internl ietor of of tringle meet ide t D. line drwn through D perpendiulr to D interet the ide t E nd the ide t F. If,, repreent ide of E i HM of nd D o EF in d the tringle EF i ioele then 9. In tringle with fixed e, the vertex move uh tht o o in. If, nd denote the length of the ide of the tringle oppoite to the ngle, nd, repetively, then + + lou of point i n ellipe d lou of point i pir of tright line 0. tright line through the vertex P of tringle PQR interet the ide QR t the point S nd the irumirle of the tringle PQR t the point T. If S i not the entre of the irumirle, then PS ST QS SR PS ST QS SR PS ST QR d PS ST QR. In tringle with fixed e, the vertex move uh tht If, o o in. nd denote the length of the ide of the tringle oppoite to the ngle, nd, repetively, then + + lou of point i n ellipe d lou of point i pir of tright line. If the ngle, nd of tringle re in n rithmeti progreion nd if, nd denote the length of the ide oppoite to, nd repetively, then the vlue of the expreion in in i. Let PQR e tringle of re with 7, nd d 5, where, nd re the length of the
23 in P in P ide of the tringle oppoite to the ngle t P, Q nd R repetively. Then in P in P 5 d 5 equl. In tringle the um of two ide i x nd the produt of the me two ide i y. If x y, where i the third ide of the tringle, then the rtio of the in-rdiu to the irum-rdiu of the tringle i y y y y x x x x x d x DE QUESTIONS. In : : : 5 :. Then + + i equl to d. If the perimeter of tringle i 6 time the.m. of the ine of it ngle nd the ide i, then ngle i d 0. In tringle,, +, 60. Then the ide 6 d none of thee. The ngle of elevtion of top of tower from point on the ground i 0 nd it i 60 when it i viewed from point loted 0 m wy from the initil point towrd the tower. The height of the tower i d 0 IEEE/JEE-MINS QUESTIONS. The ide of tringle re in, o nd in o for ome 0 < <. Then the gretet ngle of the tringle i. If in, the ltitude from the vertie,, on oppoite ide re in H.P., then in, in, in re in rithmeti Geometri Progreion H.P. G.P. d.p.. In tringle, let. If r i the inrdiu nd R i the irumrdiu of the tringle, then r + R equl d +
24 . The um of the rdii of inried nd irumried irle for n n ided regulr polygon of ide, i ot n ot n ot n d ot n 5. The upper / th portion of vertile pole utend n ngle tn /5 t point in the horizontl plne through it foot nd t ditne 0 m from the foot. 80 m 0 m 0 m d 60 m 6. In tringle, medin D nd E re drwn. If D, D /6 nd E /, then the re of the i d 7. If in o + o, then the ide, nd tify + re in. P. re in G. P. d re in H. P. 8. peron tnding on the nk of river oerve tht the ngle of elevtion of the top of tree on the oppoite nk of the river i 60 nd when he retire 0 meter wy from the tree the ngle of elevtion eome 0. The redth of the river i 0 m 60 m 0 m d 0 m 9. i vertil pole with t the ground level nd t the top. mn find tht the ngel of elevtion of the point from ertin point on the ground i 60. He move wy from the pole long the line to point D uh tht D 7 m. From D the ngle of elevtion of the point i 5. Then the height of the pole i m m m d 7 m 0. For regulr polygon, let r nd R e the rdii of the inried nd the irumried irle. fle ttement mong the following i There i regulr polygon with There i regulr polygon with r R There i regulr polygon with r R r R d There i regulr polygon with r R. Let T n e the numer of ll poile tringle formed y joining vertie of n n-ided regulr polygon. If T n + T n 0, then the vlue of n i d 5. ird i itting on the top of vertil pole 0 m high nd it elevtion from point O on the ground i
25 5. It flie off horizontlly tright wy from the point O. fter one eond, the elevtion of the ird from O i redued to 0. Then the peed in m/ of the ird i d 0
26 SI LEVEL SSIGNMENT. If,, 5, find r nd R.. In n equilterl tringle, find the reltion etween the in-rdiu nd the irum-rdiu.. Solve the tringle, if, 6,.. If 5, 7 nd in, olve the tringle, if poile. 5. If 0º, 8 nd 6, find. 6. The ngle of tringle re in the rtio : : 7, find the rtio of it ide. 7. In, prove tht : o o o o o o o o o. 8. Prove tht + {ot / + ot / } ot /. 9. Prove tht in + in + in With uul nottion, if in tringle, then prove tht o 7 o o In tringle, prove tht tn tn tn.. irle of rdiu,, 5 touhe externlly. Find the ditne from point of ontt to interetion point of tngent.. If i the re of tringle with ide length,, then how tht.. In ny tringle, prove tht ot ot ot ot ot ot. 5. Let e tringle with inentre I nd rdiu of inirle r. Let D, E, F e the feet of the perpendiulr from I to the ide, nd repetively. If r, r, r re the rdii of irle inried in the qudrilterl FIE, DIF nd DIE repetively, Prove tht r r r r r r r r r r rr. r r r r r r
27 DVNED LEVEL SSIGNMENT. If in tringle in, prove tht it i either right ngled or n ioele tringle. in. If i lene nd o + o in / then prove tht,, re in.p.. Two ide of the tringle re of length 6 nd nd the ngle oppoite to mller ide i 0º. How mny uh tringle re poile? Find the length of their third ide nd re.. If in tringle, nd D i medin then prove tht : D In tringle, if +, prove tht : ot ot. 6. D i the mid point of in tringle. If D i perpendiulr to, prove tht o o. 7. If p, p, p e the ltitude of tringle from the vertie,, repetively nd e the re of the tringle, prove tht p p p o 8. The ide of tringle re three oneutive nturl numer nd it lrget ngle i twie the mllet one. Determine the ide of the tringle. 9. In tringle, prove tht o o tn tn e. o o o o 0. Prove tht the rdiu of the irle ping through the entre of the inried irle of the tringle nd through the end point of the e i e.. Perpendiulr re drwn from the vertie,, of n ute ngled tringle on the oppoite ide, nd produed to meet the irumirle of the tringle. If thee produed prt e,, repetively, how tht tn + tn + tn.
28 . If in tringle 8R + +, prove tht the tringle i right ngled.. In tringle of e, the rtio of the other two ide i r <. Show tht the ltitude of the tringle i le thn or equl to r r.. Let e tringle hving O nd I it irumentre nd inentre repetively if R nd r e the irum rdiu nd the inrdiu repetively, then prove tht IO R Rr. Further how tht the tringle IO i right ngled tringle if nd only if i the.m. of nd. 5. If nd re the ltitude of the from the vertie, nd repetively then how tht ot + ot + ot
29 NSWERS Ojetive ignment.. d.. d d d d 0..,..,., 5.,d 6.,, 7., 8.,, 9., 0., Miellneou ignment ; -r; -p; D-q 9. -r; -q; -p; D Previou Yer Quetion IIT-JEE/JEE-DVNE ,, d d ,d.,d 5. d ,,,d 9., 0. d.,. d.. DE QUESTIONS.... d
30 MINS QUESTIONS.. d. d. d d d. d. d i Level ignment. r, R r R 5.,,. No tringle n e formed ::. 5 dvned Level ignment. Side re, 8., 5 nd 6.
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