Question

In an equilateral triangle, the in-radius, circum-radius and one of the ex-radii are in the ratio ______.

Options

• 2:3:5

• 1:2:3

• 1:3:7

• 3:7:9

Answer 1

In an equilateral triangle, the in-radius, circum-radius and one of the ex-radii are in the ratio 1:2:3.

Explanation:

In equilateral triangle

Δ = `sqrt(3)/4a^2`

R = `(abc)/(4Δ) = a^3/(4.sqrt(3)/4a^2) = a/sqrt(3)`

r = `Δ/s = (sqrt(3)/4a^2)/(3/2a) = a/(2sqrt(3))`

r₁ = `Δ/(s - a) = (sqrt(3)/4a^2)/(1/2a) = (sqrt(3)a)/2 = (3a)/(2sqrt(3))` 

⇒ r:R:r₁ = 1:2:3