$\triangle ABC$ is a right-angled triangle with $AB=12cm$ and $AC=13cm$. A circle with centre $O$ has been inscribed inside the triangle. The value of $x$, the radius of the inscribed circle is

Using the Pythagorean theorem for the right triangle, the third side BC is the square root of (13 squared minus 12 squared), which equals the square root of 25, or 5 cm. The radius of an inscribed circle in a right triangle is found by the formula r equals the area divided by the semi-perimeter. The area is half of 12 times 5, which is 30 square cm, and the semi-perimeter is (12 plus 13 plus 5) divided by 2, which is 15 cm. Dividing the area 30 by the semi-perimeter 15 gives the radius x equals 2 cm.

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