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.
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