CD – the height of the right-angled triangle ABC, drawn from the vertex of the right angle. AD = 3cm, BD = 24cm. Find AC.

The height drawn to the hypotenuse in a right-angled triangle divides it into two similar triangles. So triangles ACD and BCD are similar. This implies:

CD / AD = BD / CD;

CD² = AD * BD = 3 * 24 = 72 (cm²);

By the Pythagorean theorem from the triangle ACD:

AC² = AD² + CD²;

AC² = 9 +72 = 75 (cm²);

AC = √75 = 5 √3 (cm);