In a right-angled triangle, the sum of the legs is 14, the radius of the inscribed circle is 2, find the hypotenuse.

Let’s stand the radii OK, ОМ and ОН.

The quadrangle BKON is a square, then the segments KB = BH = R = 2 cm.

Let side AB = b, BC = a, AB = c.

Then, by the property of tangents drawn from one point: AK = AM = c – R.

CM = CH = a – R.

Then AC = AM + CM = c – R + a – R = (a + c) – 2 * R.

By condition, (a + c) = 14 cm, then AC = 14 – 2 * 2 = 10 cm.

Answer: The length of the hypotenuse is 10 cm.