# In a right-angled triangle, the height is drawn from a right angle to the hypotenuse, find the radius of the inscribed circle

In a right-angled triangle, the height is drawn from a right angle to the hypotenuse, find the radius of the inscribed circle, if the hypotenuse is divided into 2 segments, one of which is 14.4, and the larger one is 25.6. Find the radius of the inscribed circle.

The height CH divides the triangle ABC into two right-angled triangles, ACH and BCH.

Let us prove the similarity of triangles ACH and BCH.

Let the angle CAH = X0, then the angle ACH = (90 – X) 0. Angle АСВ = 90, then angle ВСН = (90 – (90 – X)) = X0.

Then the right-angled triangle ACH is similar to the right-angled triangle BCH in the stroma angle.

In such triangles, the ratio of similar sides is equal, then: AH / CH = CH / BH.

CH ^ 2 = AH * BH = 25.6 * 14.4 = 368.64.

CH = 19.2 cm.

By the Pythagorean theorem, we define the legs of the triangle ABC.

AC ^ 2 = AH ^ 2 + CH ^ 2 = 655.36 + 368.64 = 1024.

AC = 32 cm.

BC ^ 2 = BH ^ 2 + CH ^ 2 = 207.36 + 368.64 = 576.

BC = 24 cm.

Determine the radius of the inscribed circle.

R = (AC + BC – AB) / 2 = (32 + 24 – 40) / 2 = 8 cm. 