# Puzzles

## Semi circle in a triangle

This right-angled triangle above has sides of lengths 12cm, 5cm and 13cm. The diameter of the semicircle lies on the 12cm side and the 13cm side is a tangent to the circle. What is the radius of the semi circle?

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Label the triangle as follows:

EC and BC are both tangents to the circle so FEA is a right angle and the lengths EC and BC are equal, so the length of EC is 5. Let *r* be the radius of the circle.

Using Pythagoras' Theorem in triangle FEA:

$$8^2+r^2=(12-r)^2$$
$$64+r^2=144-24r+r^2$$
$$24r=80$$
$$r=\frac{80}{24}=\frac{10}{3}$$

#### Extension

What is the radius of the circle whose diameter lies on the 5cm side and to which the 12cm and 13cm sides are tangents?

What is the radius of the circle whose diameter lies on the 13cm side and to which the 12cm and 5cm sides are tangents?

**Additional observation**

For each pair of semi circles, draw a straight line between the two points where the semi circles intersect. These lines all meet at a point.