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# Matter of Scale

**What are the lengths of these new triangles? **

Draw out a copy of them and indicate what the lengths and the angles are in each.
*We are very grateful to the Heilbronn Institute for Mathematical Research for their generous support for the development of this resource.*
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Age 14 to 16

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.*

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

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?

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