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.