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

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

Enlargements
Rotate these triangles and fit them together to make a third triangle:
 
 
Enlargements fitted together

Prove that the resulting triangle is 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.