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?