A circle is inscribed in a right-angled triangle. The point of contact divides the hypotenuse

A circle is inscribed in a right-angled triangle. The point of contact divides the hypotenuse into 3 cm and 2 cm segments, find the radius of this circle.

It is known from the property of tangency to the circle that the lengths of the segments drawn from the vertex of the triangle to the tangency points are equal. Let the radius of the circle be x, then one leg will be x + 2, and the other x + 3. Our hypotenuse is 5. The square of the hypotenuse of a right triangle is equal to the sum of the squares of its legs. Let’s make the equation:

(x + 2) ^ 2 + (x +3) ^ 2 = 25;
x ^ 2 + 4x + 4+ x ^ 2 + 6x + 9 = 25;
2x ^ 2 + 10x – 12 = 0;
Divide the left and right sides of the equation by 2;
x ^ 2 + 5x – 6 = 0;
x1 = -6 – does not satisfy;
x2 = 1 – satisfies;