# 2021 JMPSC Accuracy Problems/Problem 12

## Contents

## Problem

A rectangle with base and height is inscribed in an equilateral triangle. Another rectangle with height is also inscribed in the triangle. The base of the second rectangle can be written as a fully simplified fraction such that Find .

## Solution

We are given , from which in rectangle we can conclude . Since , we have

Since is parallel to and , we have that by corresponding angles. Similarly, and it follows that is a right triangle.

Since the side opposite the angle in is , we use our ratios to find that In rectangle , we also have Analogously, we find that Since we are looking for the base of the horizontal rectangle and we are given we have This gives us an answer of ~ samrocksnature

## Solution 2

Since the angles of an equilateral triangle are , we have by similar triangles that the length of the segments of the "small" rectangles are , meaning the answer is ~Geometry285

## See also

The problems on this page are copyrighted by the Junior Mathematicians' Problem Solving Competition.