# The legs of a right-angled triangle are 10 and 24 cm. Find the segments into which the point of tangency

**The legs of a right-angled triangle are 10 and 24 cm. Find the segments into which the point of tangency of the inscribed circle divides the hypotenuse.**

Let us designate this right-angled triangle ABC, leg BC = 10 cm, AC = 24 cm, point O – the center of the inscribed circle, OK – radius to leg AC, OM – radius to leg BC, OH – radius to hypotenuse AB.

By the Pythagorean theorem, we find the hypotenuse AB:

AB = √ (BC² + AC²) = √ (100 + 576) = √676 = 26 (cm).

Let’s designate the side of the square of the OКCM x cm, then:

MВ = BC – CM = 10 – x;

KA = AC – СK = 24 – x.

The property of tangents drawn from one point allows us to write equality:

ВН = MВ = 10 – x;

AH = KA = 24 – x.

Together, these segments give us the hypotenuse AB:

(10 – x) + (24 – x) = 26

-2x = – 8

x = 4

BH = 10 – 4 = 6 (cm);

AH = 24 – 4 = 20 (cm).

Answer: 6 cm and 20 cm.