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?
Age
11 to 14
Challenge level
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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