Question

The ratio of the perimeter of an equilateral triangle having an altitude equal to the radius of a circle, to the perimeter of an equilateral triangle inscribed in the circle is :

• A. $1:2$
• B. $1:3$
• C. $1:\sqrt{3}$
• D. $\sqrt{3}:2$
• E. $2:3$

Answer 1

The correct option is E $2:3$

Let radius of the circle = r

Height of the first equilateral triangle = $\frac{\sqrt{3}}{2}{a}_1$

$⟹\frac{\sqrt{3}}{2}{a}_1=r⟹a_1=\frac{2r}{\sqrt{3}}$

Let $a_2$ be side of equilateral triangle inscribed in circle.

$r\cos 30°=\frac{a_2}{2}⟹a_2=\sqrt{3}r\therefore \frac{perimeteroftheequilateraltriangle1}{perimeteroftheequilateraltriangle2}=\frac{3a_1}{3a_2}=\frac{\frac{2}{\sqrt{3}}}{\sqrt{3}}=2:3$