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Poly-puzzle

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3
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This rectangle is cut into five pieces which fit exactly into a triangular outline and also into a square outline where the triangle, the rectangle and the square have equal areas.

Can you cut up any triangle, using straight cuts, and rearrange the pieces to make a rectangle and a square?
Start by just investigating triangles which have a height of less than twice the base but greater than half the base. Make your own puzzle from card by cutting the triangle into a small number of pieces that can be re-arranged into a rectangle and into a square with the same area.

Suppose you have two triangles with the same area. Do you think you can cut one into pieces which can be re-arranged to form the other?