In a right-angled triangle, the height dropped from the vertex of the right angle divides the hypotenuse into segments

In a right-angled triangle, the height dropped from the vertex of the right angle divides the hypotenuse into segments of length 9 and 16. Find the radius of the circle inscribed in the triangle.

1. A circle is inscribed in triangle ABC. R is its radius. ∠А = 90 °. AE – height. BE = 9 cm.

CE = 16cm.

2. The height AE is drawn from the vertex ∠A, equal to 90 °. Therefore, in accordance with the properties

a right-angled triangle, calculated by the formula:

AE = √BE x CE = √9 x 16 = √144 = 12 cm.

3. AC = √AE² + CE² = √12² + 16² = √144 + 256 = √400 = 20 cm.

4. AB = BE² + AE² = √9² + 12² = √81 + 144 = √225 = 15 cm.

5. BC = BE + CE = 9 + 16 = 25 cm.

6.R = (AB + AC – BC): 2 = (15 + 20 – 25): 2 = 5 cm.