In triangle ABC , I is the incentre. If angles BIC:CIA:AIB = 5 : 6 : 7 ,Then the value of angles A : B: C is:

Asked by Kusum and Sanjeet | 11th Nov, 2014, 08:08: PM

Expert Answer:

Consider the following figure.
A B C space i s space t h e space g i v e n space t r i a n g l e space a n d space I space i s space t h e space i n c e n t r e space o f space t h e space c i r c l e. A l s o space g i v e n space t h a t space angle B I C : angle C I A : angle A I B equals 5 : 6 : 7 F r o m space t h e space d i a g r a m space i t space i s space c l e a r space t h a t comma 5 x plus 6 x plus 7 x equals 360 degree rightwards double arrow 18 x equals 360 degree rightwards double arrow x equals fraction numerator 360 degree over denominator 18 end fraction rightwards double arrow x equals 20 degree T h u s comma space space angle B I C equals 5 cross times 20 degree equals 100 degree angle C I A equals 6 cross times 20 degree equals 120 degree angle A I B equals 7 cross times 20 degree equals 140 degree W e space n e e d space t o space f i n d space t h e space a n g l e s space angle A comma space angle B space a n d space angle C. L e t space angle A equals a ; space angle B equals b ; angle C equals c S i n c e space i n c e n t r e space i s space t h e space i n t e r s e c t i n g space p o i n t space o f space a l l space t h e space a n g l e space b i s e c t o r s space f r o m space t h e space v e r t i c e s comma space w e space h a v e comma angle B A I equals a over 2 ; space angle I B A equals b over 2 ; space I B C equals b over 2 ; space angle I C B equals c over 2 ; space angle I C A equals c over 2 ; space angle I A C equals a over 2 C o n s i d e r space t h e space t r i a n g l e space A I B B y space a n g l e space s u m space p r o p e r t y comma space w e space h a v e comma angle A I B plus angle B A I plus angle I B A equals 180 degree rightwards double arrow 140 degree plus a over 2 plus b over 2 equals 180 degree rightwards double arrow a over 2 plus b over 2 equals 180 degree minus 140 degree rightwards double arrow fraction numerator a plus b over denominator 2 end fraction equals 40 degree rightwards double arrow a plus b equals 2 cross times 40 degree rightwards double arrow a plus b equals 80 degree S i m i l a r l y comma space C o n s i d e r space t h e space t r i a n g l e space B I C B y space a n g l e space s u m space p r o p e r t y comma space w e space h a v e comma angle I B C plus angle B I C plus angle I C B equals 180 degree rightwards double arrow b over 2 plus 100 degree plus c over 2 plus equals 180 degree rightwards double arrow b over 2 plus c over 2 equals 180 degree minus 100 degree rightwards double arrow fraction numerator b plus c over denominator 2 end fraction equals 80 degree rightwards double arrow b plus c equals 2 cross times 80 degree rightwards double arrow b plus c equals 160 degree N o w comma space C o n s i d e r space t h e space t r i a n g l e space C I A B y space a n g l e space s u m space p r o p e r t y comma space w e space h a v e comma angle I C A plus angle C I A plus angle I A C equals 180 degree rightwards double arrow c over 2 plus 120 degree plus a over 2 plus equals 180 degree rightwards double arrow c over 2 plus a over 2 equals 180 degree minus 120 degree rightwards double arrow fraction numerator c plus a over denominator 2 end fraction equals 60 degree rightwards double arrow c plus a equals 2 cross times 60 degree rightwards double arrow c plus a equals 120 degree T h u s comma space w e space h a v e comma space t h r e e space e q u a t i o n s comma a plus b equals 80 degree... left parenthesis 1 right parenthesis b plus c equals 160 degree... left parenthesis 2 right parenthesis c plus a equals 120 degree... left parenthesis 3 right parenthesis F r o m space t h e space t r i a n g l e space A B C comma space b y space a n g l e space s u m space p r o p e r t y comma space w e space h a v e comma a plus b plus c equals 180 degree... left parenthesis 4 right parenthesis U sin g space e q u a t i o n s space left parenthesis 1 right parenthesis comma space left parenthesis 2 right parenthesis comma space left parenthesis 3 right parenthesis space a n d space left parenthesis 4 right parenthesis comma space w e space h a v e comma a equals 20 degree comma space b equals 60 degree space a n d space c equals 100 degree

Answered by Vimala Ramamurthy | 12th Nov, 2014, 10:00: AM