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A square is drawn in side a triangle with sides 3, 4 and 5 such that one corner of the square touches the side 3 of the triangle, another corner touches the side 4 of the triangle, and the base of the square rests on the longest side of the triangle. What is the side of the square?
Correct Answer: 1.62
The altitude to the side 5 is $$h = \frac{3 \times 4}{5} = \frac{12}{5}$$. The top side of the square cuts off a smaller similar triangle of height $$h - s$$ and base $$s$$, so $$\frac{s}{5} = \frac{h - s}{h}$$, which gives $$s = \frac{5h}{5 + h} = \frac{12}{37/5} = \frac{60}{37} \approx 1.62$$.
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