SHORT REVISION SOLUTIONS OF TRIANGLE

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1 FREE Download Study Package fom website: wwwtekoclassescom SHORT REVISION SOLUTIONS OF TRINGLE I SINE FORMUL : In any tiangle BC, II COSINE FORMUL : (i) b + c a bc a b c sin sinb sin C o a² b² + c² bc c + a b a + b c (ii) B (iii) C ca ab III PROJECTION FORMUL : (i) a b C + c B (ii) b c + a C (iii) c a B + b B C bc IV NPIER S NLOGY TNGENT RULE : (i) tan cot b+ c C ca B B ab C (ii) tan cot (iii) tan cot c+ a a+ b V TRIGONOMETRIC FUNCTIONS OF HLF NGLES : ( sb)(sc) (i) sin bc s(sa) (ii) bc (sb)(sc) (iii) tan s(sa ) B ( sc)(sa) ; sin ca B s(sb) ; ca s(sa) (iv) ea of tiangle s(s a)(sb)(sc) VI M N RULE : In any tiangle, (m + n) cot θ m cot α n cot β n cot B mcot C VII VIII C ( sa)(sb) ; sin ab ; C s(sc) ab ab sin C bc sin ca sin B aea of tiangle BC a b c sin sin B sin C R Note that R a b c 4 Radius of the incicle is given by: (a) s whee s a + b + c sin sin (c) a B C a + b+ c whee s & aea of tiangle ; Whee R is the adius of cicumcicle & is aea of tiangle (b) (s a) tan (s b) tan B (s c) tan C & so on (d) 4R sin sin B sin C IX Radius of the Ex cicles, & ae given by : (a) ; s a ; s b s (c) a B C c (b) s tan ; s tan B ; s tan C & so on (d) 4 R sin B C 4 R sin B C ; 4 R sin C B X LENGTH OF NGLE BISECTOR & MEDINS : If m a and β a ae the lengths of a median and an angle bisecto fom the angle then, ; Page : of PROPERTIES OF TRINGLES TEKO CLSSES, HOD MTHS : SUHG R KRIY (S R K Si) PH: , , BHOPL

2 FREE Download Study Package fom website: wwwtekoclassescom m a b + c a and β a bc b + c Note that m + m + m 4 (a + b + c ) a b c XI ORTHOCENTRE ND PEDL TRINGLE : The tiangle KLM which is fomed by joining the feet of the altitudes is called the pedal tiangle the distances of the othocente fom the angula points of the BC ae R, R B and R C the distances of P fom sides ae R B C, R C and R B the sides of the pedal tiangle ae a ( R sin ), b B ( R sin B) and c C ( R sin C) and its angles ae, B and C cicumadii of the tiangles PBC, PC, PB and BC ae equal XII EXCENTRL TRINGLE : The tiangle fomed by joining the thee excentes I, I and I of BC is called the excental o excentic tiangle Note that : Incente I of BC is the othocente of the excental I I I BC is the pedal tiangle of the I I I the sides of the excental tiangle ae 4 R, 4 R B and 4 R C and its angles ae, B and C I I 4 R sin ; I I 4 R sin B ; I I 4 R sin C XIII THE DISTNCES BETWEEN THE SPECIL POINTS : (a) The distance between cicumcente and othocente is R 8 B C (b) The distance between cicumcente and incente is R R (c) The distance between incente and othocente is 4R B C XIV Peimete (P) and aea () of a egula polygon of n sides inscibed in a cicle of adius ae given by P n sin and n sin n n Peimete and aea of a egula polygon of n sides cicumscibed about a given cicle of adius is given by P n tan and n tan n n EXERCISE I With usual notations, pove that in a tiangle BC: b c c a a b Q Q a cot + b cot B + c cot C (R + ) Q Q5 + + (s b) (s c) (s c) (s a) (s a) (s b) Q4 + a b abc B C C C Q6 ( + s )tan ( ) cot c B C C Q7 ( ) ( )( ) 4 R B Q8 ( + )tan +( + )tan +( + ) tan 0 c Page : of PROPERTIES OF TRINGLES TEKO CLSSES, HOD MTHS : SUHG R KRIY (S R K Si) PH: , , BHOPL

3 FREE Download Study Package fom website: wwwtekoclassescom a + b + c Q Q0 ( + ) ( + ) sin C + + Q Q bc 4R + + Q ca ab R s bc ca ab 4 Q Q5 R (sin + sin B + sin C) Q6 R R + B C s a + b + c Q7 cot + cot + cot Q8 cot + cot B + cot C 4 Q9 Given a tiangle BC with sides a 7, b 8 and c 5 If the value of the expession ( ) cot p p sin can be expessed in the fom whee p, q N and is in its lowest fom find q q the value of (p + q) Q0 If + + then pove that the tiangle is a ight angled tiangle Q If two times the squae of the diamete of the cicumcicle of a tiangle is equal to the sum of the squaes of its sides then pove that the tiangle is ight angled Q In acute angled tiangle BC, a semicicle with adius a is constucted with its base on BC and tangent to the othe two sides b and c ae defined similaly If is the adius of the incicle of tiangle BC then pove that, + + a b c Q Given a ight tiangle with 90 Let M be the mid-point of BC If the inadii of the tiangle BM and CM ae and then find the ange of Q4 If the length of the pependiculas fom the vetices of a tiangle, B, C on the opposite sides ae p, p, p then pove that + + p p p + + bc ca ab a b b c c a Q5 Pove that in a tiangle + + R b a c b a c EXERCISE II b + c c + a a + b Q With usual notation, if in a BC, ; then pove that, B C Q Fo any tiangle BC, if B C, show that C b + c b c & sin 4c c Q In a tiangle BC, BD is a median If l (BD) l and DBC Detemine the BC 4 Q4 BCD is a tapezium such that B, DC ae paallel & BC is pependicula to them If angle DB θ, BC p & CD q, show that B (p + q )sinθ p θ + q sin θ Q5 If sides a, b, c of the tiangle BC ae in P, then pove that sin ec ; sin B ec B; sin C ec C ae in HP Page : 4 of PROPERTIES OF TRINGLES TEKO CLSSES, HOD MTHS : SUHG R KRIY (S R K Si) PH: , , BHOPL

4 FREE Download Study Package fom website: wwwtekoclassescom Q6 Find the angles of a tiangle in which the altitude and a median dawn fom the same vetex divide the angle at that vetex into equal pats B C Q7 In a tiangle BC, if tan, tan, tan ae in P Show that, B, C ae in P Q8 BCD is a hombus The cicumadii of BD and CD ae 5 and 5 espectively Find the aea of hombus cot C Q9 In a tiangle BC if a + b 0c then find the value of cot + cot B Q0 The two adjacent sides of a cyclic quadilateal ae & 5 and the angle between them is 60 If the aea of the quadilateal is 4, find the emaining two sides Q If I be the incente of the tiangle BC and x, y, z be the cicum adii of the tiangles IBC, IC & IB, show that 4R R (x + y + z ) xyz 0 Q Sides a, b, c of the tiangle BC ae in HP, then pove that ec (ec + cot ) ; ec B (ec B + cot B) & ec C (ec C + cot C) ae in P Q In a BC, (i) a b B (ii) sin B sin C C (iii) tan + tan tan 0, pove that (i) (ii) (iii) (i) Q4 The sequence a, a, a, is a geometic sequence The sequence b, b, b, is a geometic sequence b ; b ; a 4 8 and a If the aea of the tiangle with sides lengths a, a and a can be expessed in the fom of p q whee p and q ae elatively pime, find (p + q) Q5 If p, p, p ae the altitudes of a tiangle fom the vetices, B, C & denotes the aea of the tiangle, pove that p + p p ab (a + b + c) Q6 The tiangle BC (with side lengths a, b, c as usual) satisfies log a log b + log c log (bc ) What can you say about this tiangle? Q7 With efeence to a given cicle, and B ae the aeas of the inscibed and cicumscibed egula polygons of n sides, and B ae coesponding quantities fo egula polygons of n sides Pove that () is a geometic mean between and B () B is a hamonic mean between and B Q8 The sides of a tiangle ae consecutive integes n, n + and n + and the lagest angle is twice the smallest angle Find n Q9 The tiangle BC is a ight angled tiangle, ight angle at The atio of the adius of the cicle cicumscibed n C n n to the adius of the cicle escibed to the hypotenuse is, : ( ) + Find the acute angles B & C lso find the atio of the two sides of the tiangle othe than the hypotenuse b n Page : 5 of PROPERTIES OF TRINGLES TEKO CLSSES, HOD MTHS : SUHG R KRIY (S R K Si) PH: , , BHOPL

5 FREE Download Study Package fom website: wwwtekoclassescom Q0 BC is a tiangle Cicles with adii as shown ae dawn inside the tiangle each touching two sides and the incicle Find the adius of the incicle of the BC Q Line l is a tangent to a unit cicle S at a point P Point and the cicle S ae on the same side of l, and the distance fom to l is Two tangents fom point intesect line l at the point B and C espectively Find the value of (PB)(PC) Q Let BC be an acute tiangle with othocente H D, E, F ae the feet of the pependiculas fom, B, and C on the opposite sides lso R is the cicumadius of the tiangle BC Given (H)(BH)(CH) and (H) + (BH) + (CH) 7 Find (a) the atio, (b) the poduct (HD)(HE)(HF) (c) the value of R EXERCISE III Q The adii,, of escibed cicles of a tiangle BC ae in hamonic pogession If its aea is 4 sq cm and its peimete is 4 cm, find the lengths of its sides [REE '99, 6] Q(a) In a tiangle BC, Let C If ' ' is the inadius and ' R ' is the cicumadius of the tiangle, then ( + R) is equal to: () a + b b + c c + a (D) a + b + c (b) In a tiangle BC, a c sin ( B + C) () a + b c c + a b b c a (D) c a b [JEE '000 (Sceening) + ] Q Let BC be a tiangle with incente ' I ' and inadius ' ' Let D, E, F be the feet of the pependiculas fom I to the sides BC, C & B espectively If, & ae the adii of cicles inscibed in the quadilateals FIE, BDIF & CEID espectively, pove that + + )( )( ) [JEE '000, 7] ( ( a + b + c) abc Q4 If is the aea of a tiangle with side lengths a, b, c, then show that: < 4 lso show that equality occus in the above inequality if and only if a b c [JEE ' 00] Q5 Which of the following pieces of data does NOT uniquely detemine an acute angled tiangle BC (R being the adius of the cicumcicle)? () a, sin, sinb a, b, c a, sinb, R (D) a, sin, R [ JEE ' 00 (Sc), ] Q6 If I n is the aea of n sided egula polygon inscibed in a cicle of unit adius and O n be the aea of the polygon cicumscibing the given cicle, pove that I n O n + I n Q7 The atio of the sides of a tiangle BC is : : The atio : B : C is n [JEE 00, Mains, 4 out of 60] () : 5 : : : : : (D) : : [JEE 004 (Sceening)] Q8(a) In BC, a, b, c ae the lengths of its sides and, B, C ae the angles of tiangle BC The coect elation is B C B C () ( b c)sin a (b c) a sin Page : 6 of PROPERTIES OF TRINGLES TEKO CLSSES, HOD MTHS : SUHG R KRIY (S R K Si) PH: , , BHOPL

6 FREE Download Study Package fom website: wwwtekoclassescom B + C B + C ( b + c)sin a (D) (b c) a sin [JEE 005 (Sceening)] (b) Cicles with adii, 4 and 5 touch each othe extenally if P is the point of intesection of tangents to these cicles at thei points of contact Find the distance of P fom the points of contact [JEE 005 (Mains), ] Q9(a) Given an isosceles tiangle, whose one angle is 0 and adius of its incicle is tiangle in sq units is Then the aea of () (D) 4 [JEE 006, ] (b) Intenal bisecto of of a tiangle BC meets side BC at D line dawn though D pependicula to D intesects the side C at E and the side B at F If a, b, c epesent sides of BC then () E is HM of b and c D EF 4bc b + c sin bc b + c (D) the tiangle EF is isosceles [JEE 006, 5] Q0 Let BC and BC be two non-conguent tiangles with sides B 4, C C and angle B 0 The absolute value of the diffeence between the aeas of these tiangles is [JEE 009, 5] Q9 07 Q, EXERCISE I EXERCISE II Q 0 Q6 /6, /, / Q8 400 Q9 50 Q0 cms & cms Q4 9 Q6 tiangle is isosceles Q8 4 Q9 B 5 ; C ; b c + 9 Q0 Q Q (a), (b), (c) 4R 8R EXERCISE III Q 6, 8, 0 cms Q (a), (b) B Q5 D Q7 D Q8 (a) B; (b) 5 Q9 (a) C, (b), B, C, D Q0 4 P T O Page : 7 of PROPERTIES OF TRINGLES TEKO CLSSES, HOD MTHS : SUHG R KRIY (S R K Si) PH: , , BHOPL

7 FREE Download Study Package fom website: wwwtekoclassescom Pat : () Only one coect option In a tiangle BC, (a + b + c) (b + c a) k b c, if : () k < 0 k > 6 0 < k < 4 (D) k > 4 9 In a BC,, b c cm and a (BC) cm Then a is () 6 cm 9 cm 8 cm (D) none of these b c If R denotes cicumadius, then in BC, is equal to a R () (B C) sin (B C) B C (D) none of these 4 If the adius of the cicumcicle of an isosceles tiangle PQR is equal to PQ ( PR), then the angle P is () (D) 6 a + bb + cc 5 In a BC, the value of is equal to: a + b + c () R R 6 In a ight angled tiangle R is equal to () s + s R s (D) R 7 In a BC, the inadius and thee exadii ae,, and espectively In usual notations the value of is equal to abc () (D) none of these 4R 8 In a tiangle if > >, then () a > b > c a < b < c a > b and b < c (D) a < b and b > c 9 With usual notation in a BC KR + + +, a b c whee 'K' has the value equal to: () 6 64 (D) 8 0 The poduct of the aithmetic mean of the lengths of the sides of a tiangle and hamonic mean of the lengths of the altitudes of the tiangle is equal to: () (D) 4 In a tiangle BC, ight angled at B, the inadius is: B + BC C B + C BC B + BC + C () (D) None The distance between the middle point of BC and the foot of the pependicula fom is : () a + b a + c b c b a + c bc (D) s + a (D) none of these In a tiangle BC, B 60 and C 45 Let D divides BC intenally in the atio :, then, () 6 (D) sin BD sin CD 4 Let f, g, h be the lengths of the pependiculas fom the cicumcente of the BC on the sides a, b and a b c abc c espectively If + + λ then the value of λ is: f g h fgh () /4 / (D) 5 tiangle is inscibed in a cicle The vetices of the tiangle divide the cicle into thee acs of length, 4 and 5 units Then aea of the tiangle is equal to: Page : 8 of PROPERTIES OF TRINGLES TEKO CLSSES, HOD MTHS : SUHG R KRIY (S R K Si) PH: , , BHOPL

8 FREE Download Study Package fom website: wwwtekoclassescom 9 ( + () ) 9 ( ) 9 ( + ) 9 ( ) (D) 6 If in a tiangle BC, the line joining the cicumcente and incente is paallel to BC, then B + C is equal to: () 0 (D) none of these 7 If the incicle of the BC touches its sides espectively at L, M and N and if x, y, z be the cicumadii of the tiangles MIN, NIL and LIM whee I is the incente then the poduct xyz is equal to: () R R R (D) R 8 If in a BC, (), then the value of tan B C tan + tan is equal to : (D) None of these 9 In any BC, then minimum value of is equal to () 9 7 (D) None of these 0 In a acute angled tiangle BC, P is the altitude Cicle dawn with P as its diamete cuts the sides B and C at D and E espectively, then length DE is equal to () (D) R R 4R R, BB and CC ae the medians of tiangle BC whose centoid is G If the concyclic, then points, C, G and B ae () b a + c c a + b a b + c (D) None of these In a BC, a, b, ae given and c, c ae two values of the thid side c The sum of the aeas of two tiangles with sides a, b, c and a, b, c is () b sin a sin b sin (D) none of these In a tiangle BC, let C If is the inadius and R is the cicumadius of the tiangle, then ( + R) is equal to () a + b c b + c c + a (D) a + b + c [IIT - 000] 4 Which of the following pieces of data does NOT uniquely detemine an acute - angled tiangle BC (R being the adius of the cicumcicle )? [IIT - 00] () a, sin, sin B a, b, c a, sin B, R (D) a, sin, R 5 If the angles of a tiangle ae in the atio 4 : :, then the atio of the longest side to the peimete is () : ( + ) : 6 : + (D) : [IIT - 00] 6 The sides of a tiangle ae in the atio : :, then the angle of the tiangle ae in the atio [IIT - 004] () : : 5 : : 4 : : (D) : : 7 In an equilateal tiangle, coincs of adii unit each ae kept so that they touche each othe and also the sides of the tiangle ea of the tiangle is [IIT - 005] () (D) + P + PB + PC + PD 8 If P is a point on C and Q is a point on C, then equals Q + QB + QC + QD () / /4 5/6 (D) 7/8 9 cicle C touches a line L and cicle C extenally If C and C ae on the same side of the line L, then locus of the cente of cicle C is () an ellipse a cicle a paabola (D) a hypebola 7 4 Page : 9 of PROPERTIES OF TRINGLES TEKO CLSSES, HOD MTHS : SUHG R KRIY (S R K Si) PH: , , BHOPL

9 FREE Download Study Package fom website: wwwtekoclassescom 0 Let l be a line though and paallel to BD point S moves such that its distance fom the line BD and the vetex ae equal If the locus of S meets C in, and l in and, then aea of is () 05 (unit) 075 (unit) (unit) (D) (/) (unit) Pat : May have moe than one options coect In a BC, following elations hold good In which case(s) the tiangle is a ight angled tiangle? () + a + b + c 8 R s (D) R In a tiangle BC, with usual notations the length of the bisecto of angle is : bc sin bc abc ec () (D) ec b + c b + c R(b + c) b + c D, BE and CF ae the pependiculas fom the angula points of a BC upon the opposite sides, then : Peimete of DEF () ea of DEF B C Peimete of BC R ea of EF (D) Cicum adius of DEF R 4 The poduct of the distances of the incente fom the ( angula points of a BC is: abc ) R ( abc ) () 4 R 4 R (D) s s 5 In a tiangle BC, points D and E ae taken on side BC such that BD DE EC If angle DE angle ED θ, then: () tanθ tan B tanθ tanc 6tanθ tan tan θ 9 (D) angle B angle C 6 With usual notation, in a BC the value of Π ( ) can be simplified as: () abc Π tan 4 R ( a bc) ( + + ) R a b c (D) 4 R + C If in a tiangle BC, sin B, pove that the tiangle BC is eithe isosceles o ight + B sinc angled In a tiangle BC, if a tan + b tan B (a + b) tan + B, pove that tiangle is isosceles If then pove that the tiangle is the ight tiangle 4 In a BC, C 60 & 75 If D is a point on C such that the aea of the BD is times the aea of the BCD, find the BD 5 The adii,, of escibed cicles of a tiangle BC ae in hamonic pogession If its aea is 4 sq cm and its peimete is 4 cm, find the lengths of its sides 6 BC is a tiangle D is the middle point of BC If D is pependicula to C, then pove that ( a ) c C ac 7 Two cicles, of adii a and b, cut each othe at an angle θ Pove that the length of the common chod is ab sinθ a + b + abθ 8 In the tiangle BC, lines O, OB and OC ae dawn so that the angles OB, OBC and OC ae each equal to ω, pove that (i) cot ω cot + cot B + cot C (ii) ec ω ec + ec B + ec C 9 In a plane of the given tiangle BC with sides a, b, c the points, B, C ae taken so that the BC, B C and BC ae equilateal tiangles with thei cicum adii R a, R b, R c ; inadii a, b, c & exadii a, b & c espectively Pove that; [ ( R )] a a a (i) Π a : Π R a : Π a : 8: 7 (ii) Πtan The tiangle BC is a ight angled tiangle, ight angle at The atio of the adius of the cicle Page : 0 of PROPERTIES OF TRINGLES TEKO CLSSES, HOD MTHS : SUHG R KRIY (S R K Si) PH: , , BHOPL

10 cicumscibed to the adius of the cicle escibed to the hypotenuse is, : ( ) + Find the acute angles B & C lso find the atio of the two sides of the tiangle othe than the hypotenuse The tiangle BC is a ight angled tiangle, ight angle at The atio of the adius of the cicle cicumscibed to the adius of the cicle escibed to the hypotenuse is, : ( ) + Find the acute angles B & C lso find the atio of the two sides of the tiangle othe than the hypotenuse If the cicumcente of the BC lies on its incicle then pove that, + B + C Thee cicles, whose adii aea a, b and c, touch one anothe extenally and the tangents at thei points of contact meet in a point; pove that the distance of this point fom eithe of thei points of contacts abc is a + b + c Page : of PROPERTIES OF TRINGLES FREE Download Study Package fom website: wwwtekoclassescom EXERCISE # C B B 4 D 5 6 B 7 B 8 9 C 0 B B C B 7 C 8 B 9 C 0 D C 4 D B 8 B 9 C 0 C BCD CD BCD 4 BD 5 CD 6 CD EXERCISE # 4 BD 0 5 6, 8, 0 cms 5 b 0 B, C, c + 5 b B, C, c + TEKO CLSSES, HOD MTHS : SUHG R KRIY (S R K Si) PH: , , BHOPL