# In ABC, I is the incentre. If BIC : CIA : AIB= 5 : 6 : 7, then the value of A : B : C is(1) 3 : 1 : 5 (2) 1 : 3 : 5(3) 5 : 3 : 1 (4) 1 : 5 : 3

1
by nikus
are given values areas or angles ?

2014-11-17T17:41:24+05:30

### This Is a Certified Answer

Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest.
Look at the diagram.

ID, IE, IF are radii of incircle and perpendiculars on to the sides from incenter I.

Triangles AIE, AIF are similar as AI is common, IE=IF, and angles IEA = IFA = 90 deg.  Also, AE=AF as tangents from same point A to incircle.
Similarly, triangles BIF, BID are similar.  Also,  CID, and CIE are similar.

Hence, angle AIF=AIE= z.
In the quadrilateral AFIE,      angle A = 360-(90+90+FIE) = 180  - 2 z
Similarly from quadrilateral BFID, we get         angle B = 180 - 2 x
From quadrilateral CDIE,  we get    angle C = 180 - 2 y

We are given that BIC : CIA : AIB  = 5 : 6 : 7

x+y :  y+z  :  z+x  = 5 : 6 : 7

(x+y)6 = (y+z)5    =>  y + 6x = 5z
(x+y)7 = (z+x) 5   =>  2x + 7y = 5z

solving these simultaneous equations, we get 6 y = 4 x, or 3 y = 2 x
and 10 y = 5z,   z = 2y

angle A = 180 - 2 z = 180 - 4y,       angle B = 180 - 2 x = 180 - 3y
angle C = 180 - 2 y

Their sum = A+B+C = 180 =>   9 y = 360,  y = 40 deg.

angle A = 20 deg,   angle B = 60 deg     angle C = 100 deg.

A : B : C  = 1 : 3 : 5