Question

Let a, b and c be the length of sides of a triangle ABC such that `(a + b)/7 = (b + c)/8 = (c + a)/9`. If r and R are the radius of incircle and radius of circumcircle of the triangle ABC, respectively, then the value of `R/r` is equal to ______.

Options

• `5/2`

• 2

• `3/2`

• 1

Answer 1

Let a, b and c be the length of sides of a triangle ABC such that `(a + b)/7 = (b + c)/8 = (c + a)/9`. If r and R are the radius of incircle and radius of circumcircle of the triangle ABC, respectively, then the value of `R/r` is equal to `underlinebb(5/2)`.

Explanation:

Given relation is

`(a + b)/7 = (b + c)/8 = (c + a)/9` = k

a + b = 7k, b + c = 8k, a + c = 9k

Add above three equations

`\implies` a + b + c = 12k

After solving all the equations, we get

a = 4k, b = 3k, c = 5k

Now apply Heron's formula to find the area of triangle.

s = `(a + b + c)/2 = (4k + 3k + 5k)/2` = 6k

Δ = `sqrt(s(s - a)(s - b)(s - c))`

= `sqrt(6k(6k - 4k)(6k - 3k)(6k - 5k))`

= `sqrt(6k(2k)(3k)(k))`

= 6k²

Take, R = `(abc)/(4Δ) = ((4k)(5k)(3k))/(4 xx 6k^2) = (5k)/2`

Now, take r = `Δ/S = (6k^2)/(6k)` = k

Divide R by r,

`R/r = ((5k)/2)/k = 5/2`