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. One of the components of a person's success in our time is receiving modern high-quality education, mastering the knowledge, skills and abilities necessary for life in society. A person today needs to study almost all his life, mastering everything new and new, acquiring the necessary professional qualities.