Problem
GEO-B1-M03-P019 A Square in a Triangle
A square is inscribed in a triangle with base \(a\) and altitude to this base \(h\), so that one side of the square lies on the base and the two upper vertices lie on the lateral sides. Find the side of the square.
The cross-section of the triangle at height \(x\) from the base has length \(a\left(1-\frac{x}{h}\right)\). For the square, this length equals \(x\).
Let the side of the square be \(x\). The top side of the square is parallel to the base of the triangle and lies at distance \(x\) from the base. The small triangle above the square is similar to the original triangle; its altitude is \(h-x\), so the length of the top side of the square is \(a\cdot\frac{h-x}{h}\). But this length equals \(x\). Thus \(x=a\cdot\frac{h-x}{h}\), or \(xh=ah-ax\). Hence \(x(a+h)=ah\), so \(x=\frac{ah}{a+h}\).
Level 4: students must choose a variable and see the similar top triangle.