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 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 increases for $a<1$ and strictly decrease for $a>1$

Answer 1

Answer: (C)

For an equilateral triangle $ABC$ having side length $a$. If $R$ and r are radii of the circumcircle and the incircle of triangle $ABC$ respectively, then

$R=\frac{a}{2}\sec 30^{\circ }$

$=\frac{a}{2}(\frac{2}{\sqrt{3}})$

$=\frac{a}{\sqrt{3}}$



and $r=\frac{a}{2}\tan 30^{\circ }$

$=\frac{a}{2}\times \frac{1}{\sqrt{3}}$

$=\frac{a}{2\sqrt{3}}$

$\therefore \frac{R}{r}=\frac{\frac{a}{\sqrt{3}}}{\frac{a}{2\sqrt{3}}}=2$

which is independent of $a$ and it is constant.