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# Compare Areas

**Why do** this problem **?**

**Possible approach**

**Key questions**

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### Some(?) of the Parts

### Triangle Incircle Iteration

### Medallions

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Age 14 to 16

Challenge Level

- Problem
- Getting Started
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This problem brings together ideas of areas of circles and squares, the use of Pythagoras' Theorem and the property of tangents to a circle from an exernal point.

You might start with the middle diagram which is the easiest. It brings in the ratio of the sides of an isosceles right angles triangle which is again used in the other two parts.

If you know the side length of an isosceles right angled triangle how do you find the hypotenuse?

Which lengths are equal in the diagram?

Which angles are equal?

Can you use the symmetry of the diagram?

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

Keep constructing triangles in the incircle of the previous triangle. What happens?

Three circular medallions fit in a rectangular box. Can you find the radius of the largest one?