Question

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

• A. 2: 3: 5
• B. 1: 2: 3
• C. 1: 3: 7
• D. 3: 7: 9

Answer 1

Answer: (B)

We have; $\Delta =\frac{\sqrt{3}}{4}{a}^2,s=\frac{3a}{2}$
$\therefore r=\frac{\Delta }{S}=\frac{a}{2\sqrt{3}},R=\frac{abc}{4\Delta }$
$=\frac{a^3}{\sqrt{3}\cdot a^2}=\frac{a}{\sqrt{3}}$
and $r_1=\frac{\Delta }{S-a}=\frac{\sqrt{3}/4\cdot a^2}{a/2}=\frac{\sqrt{3}}{2}\cdot a$
Hence; $r:R:r_1=\frac{a}{2\sqrt{3}}:\frac{a}{\sqrt{3}}:\frac{\sqrt{3}}{2}\cdot a=1:2:3$