Let `C` be incircle of ` A B Cdot` If the tangents of lengths `t_1,t_2 and t_3` are drawn inside the given triangle parallel to sidese `a , b and c` , respectively, the `(t_1)/a+(t_2)/b+(t_3)/c` is equal to 0 (b) 1 (c) 2 (d) 3

Options

A. 0
B. 1
C. 2
D. 3

Answer

B

Explanation

`DeltaAP_(1) P_(2) ~ DeltaABC`
`rArr (t_(1))/(a) = (h-2r)/(h) = 1 -(2r)/(h)`
`rArr (t_(1))/(a) = 1 -(2Delta)/(h)`(where `Delta ~= ar (DeltaABC)`)
`rArr (t_(1))/(a) = 1 -(2(1)/(2) ah)/(sh)`
`rArr (t_(1))/(a) = 1 - (a)/(s)`
Similarly, `(t_(2))/(b) = 1 - (b)/(s) and, (t_(3))/(c) = 1 -(c)/(s)`
`:. (t_(1))/(a) + (t_(2))/(b) + (t_(3))/(c) = 3 - ((a+b+c))/(s) = 3 -2 =1`