The lengths of the tangents drawn from the vertices A, B and C to the incicle of `DeltaABC` are 5, 3 and 2, respectively. If the lengths of the parts of tangents within the triangle which are drawn parallel to the sides BC, CA and AB of the triangle to the incircle are `alpha, beta and gamm`, respectively, then the value of `[alpha + beta + gamma]` (where `[.]` respresents the greatest integer functin) is _____

Answer

6

Explanation

Given `s - a = 5, s -b =3, s-c =2`
`rArr s= 10, a = 5, b = 7, and c = 8`
Now, Length of tangent from A to incircle
= Semiperimeter of `DeltaAPQ`
`5 = ((calpha)/(a) + alpha + (b alpha)/(a))/(2)`
`rArr alpha = (10a)/(a+b+c) = (5)/(2) " " [ :' Delta APQ and DeltaABC " are similar"]`
Similarly, `beta = (6b)/(a+b+c) = (21)/(10)`
and `gamma = (4c)/(a+b+c) = (8)/(5)`
`rArr alpha + beta + gamma + (62)/(10)`