Question

In an equilateral triangle, the inradius, circumradius and one of the exradii 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, $\Delta =\frac{\sqrt{3}}{4}{a}^2,s=\frac{3a}{2}$
Inradius $r=\frac{\Delta }{s}=\frac{a}{2\sqrt{3}}$
Circumradius $R=\frac{abc}{4\Delta }=\frac{a^3}{\sqrt{3}{a}^2}=\frac{a}{\sqrt{3}}$
and exradii $r_1=\frac{\Delta }{s-a}=\frac{\sqrt{3}/4a^2}{a/2}$
$=\frac{\sqrt{3}}{2}a$
$\therefore$ Required ratio $=r:R:r_1$
$=\frac{a}{2\sqrt{3}}:\frac{a}{\sqrt{3}}:\frac{\sqrt{3}}{2}$
$a=1:2:3.$