Take any right-angled triangle with side lengths $a, b$ and $c$.

Make two enlargements of the triangle, by scale factors $a$ and $b$:

Rotate these triangles and fit them together to make a third triangle:
You can use the interactivity below to explore this problem. Choose a triangle, then click "New Triangle" to make copies of your triangle, which can be enlarged and rotated to fit together on the grey outline.
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Prove that the resulting triangle when the pieces are fitted together is always an enlargement of the original triangle. What is the scale factor of enlargement of the resulting triangle?

Use what you have discovered about the side lengths of the resulting triangle to come up with a proof of Pythagoras' Theorem.