Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 1.8 Junior Find the answer

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:
$\textbf{(A)}\ 1:2$ $\textbf{(B)}\ 1:3$ $\textbf{(C)}\ 1:\sqrt3$ $\textbf{(D)}\ \sqrt3:2$ $\textbf{(E)}\ 2:3$

Your answer

Multiple choice: answer with the letter of the option you want.

Solution

If the radius of the circle is $r$ , then the perimeter of the first triangle is $3\left(\frac{2r}{\sqrt3}\right)=2r\sqrt3$ , and the perimeter of the second is $3r\sqrt3$ . So the ratio is $\boxed{\frac23{\textbf{ (E)}}}$ .

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