Question

In an equilateral triangle, the inradius, circumradius, 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

The correct option is B 1:2:3

Given triangle is an equilateral triangle.
$A=B=C=\frac{\pi }{3}$
$r:R:r_1=4R\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}:R:4R\sin \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}$
$\Rightarrow r:R:r_1=4{\sin }^3\frac{\pi }{6}:1:\sin \frac{\pi }{6}{\cos }^2\frac{\pi }{6}=\frac{4}{8}:1:\frac{4\times 3}{2\times 4}$
$\Rightarrow r:R:r_1=1:2:3$
Ans: B