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Some(?) of the Parts

A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle

Ladder and Cube

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

At a Glance

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

Matter of Scale

Age 14 to 16
Challenge Level

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 $b$:

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.