A right-angled triangle ABC (C = 90 °) is described around a circle centered at point O. The hypotenuse AB

A right-angled triangle ABC (C = 90 °) is described around a circle centered at point O. The hypotenuse AB is divided by the point of contact D into segments AD = 3 and DB = 10. Find the circumference.

To solve the problem, we will use the property of tangents drawn from one point.
The radius of the inscribed circle is denoted by r. Let’s write down what the sides of a given right-angled triangle are equal to.
AC leg = r + 10;
Leg BC = r + 3;
Hypotenuse AB = 10 + 3 = 13.
We write down the Pythagorean theorem for the ABC triangle:
AB² = AC² + BC²
13² = (r + 10) ² + (r + 3) ²
r² + 20r + 100 + r² + 6r + 9 – 169 = 0
2r² + 26r – 60 = 0
r² + 13r – 30 = 0
By Vieta’s theorem, the roots of the equation:
r1 = 2, r2 = -15.
Negative root does not fit, circle radius is 2.
The circumference is 4π.
Answer: the circumference is 4π.