# A circle is inscribed in a right-angled triangle. The point of tangency of the inscribed circle

**A circle is inscribed in a right-angled triangle. The point of tangency of the inscribed circle with one of the legs divides into segments 6si and 5cm. Find the diameter of the circle circumscribed about a given right-angled triangle**

Let the BC leg be divided at the point of contact into segments BM = 6 cm and CM = 5 cm.

Then, by the property of tangents to the circle drawn from one point, BK = BM = 6 cm, CE = CM = 5 cm, AK = AE.

Then AB = X + 6 cm, AC = X + 5 cm, BC = 11 cm.

Let the lengths of the segments AK = AE = X cm, then by the Pythagorean theorem (X + 6) ^ 2 = (X + 5) ^ 2 + 112.

X ^ 2 + 12 * X + 36 = X ^ 2 + 10 * X + 25 + 121.

2 * X = 110.

X = 55 cm.

Then AB = 55 + 6 = 61 cm.

The radius of the circumscribed circle around a right-angled triangle is equal to half of its hypotenuse.

R = AB / 2 = 61/2 = 30.5 cm.

Answer: R = 30.5 m.