Question

Let $ABC$ be an equilateral triangle with side length $a$. Let $R$ and $r$ denote the radii of the circumcircle and the incircle of the triangle $ABC$ respectively. Then, as a function of $a$, the ratio $\frac{R}{r}$

• A. strictly increases
• B. strictly decreases
• C. remains constant
• D. strictly increasing for $a<1$ and strictly decreasing for $a>1$

Answer 1

The correct option is C remains constant

We know that for triangle $ABC$
$r=4R\sin \frac{A}{2}\cdot \sin \frac{B}{2}\cdot \sin \frac{C}{2}$

Given that the triangle $ABC$ is an equilateral triangle.
$\Rightarrow \angle A=\angle B=\angle C={60}^{\circ }\Rightarrow \frac{R}{r}=\text{constant}$