# Square in a Triangle

Weekly Problem 33 - 2006

A square is inscribed in an isoscles right angled triangle of area $x$. What is the area of the square?

A square is inscribed in an isoscles right angled triangle of area $x$. What is the area of the square?

## Problem

The diagram shows a right-angled isosceles triangle $XYZ$ which circumscribes a square $PQRS$. What is the ratio of the area of square $PQRS$ to the area of the triangle $XYZ$?

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## Getting Started

Can you make some congruent triangles using some simple
construction lines?

## Student Solutions

The diagram shows that triangle $XYZ$ may be divided into 9 congruent triangles. The square $PQRS$ is made up of 4 of these 9 triangles. Therefore, the ratio is $4:9$.

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