Challenge Level

Take any right-angled triangle with side lengths $a, b$ and $c$. For convenience, label the two acute angles $x^{\circ}$ and $y^{\circ}$.

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

Draw out a copy of them and indicate what the lengths and the angles are in each.

We can put these two triangles together to make a larger triangle.

Find the lengths and angles in this last triangle.

**Can you show that this triangle is similar to the original triangle?**

What is the scale factor of enlargement between the first and last triangles?

**Can you use your results to prove Pythagoras' Theorem?**

*You might like to explore some more proofs of Pythagoras' Theorem, and a proof of The Converse of Pythagoras' Theorem.*