The height lowered to the hypotenuse of a right-angled triangle divides it into segments

The height lowered to the hypotenuse of a right-angled triangle divides it into segments 9cm and 16cm. Find the sides of the triangle.

In a right-angled triangle ABC AB and BC – legs, AC – hypotenuse,

CH is the height drawn to the hypotenuse;

AH = 9 cm; BH = 16 cm;

AB = AH + BH = 9 + 16 = 25 (cm);

Triangles ABC and ACH are similar in terms of the equality of three angles, which means:

AC / AB = AH / AC;

AC² = AB * AH; AC² = 25 * 9;

AC = √ (25 * 9); AC = 5 * 3;

AC = 15 cm;

By the Pythagorean theorem:

CB² = AB² – AC²; CB² = 625 – 225 = 400 (cm²);

СB = √400 = 20 cm.