# The base of an isosceles acute-angled triangle is 48, and the radius of the circumscribed circle around it is 25.

The base of an isosceles acute-angled triangle is 48, and the radius of the circumscribed circle around it is 25. Find the distance between the centers of the inscribed and circumscribed circles of the triangle.

The radius of the circumscribed circle is: R = AC / 2 * SinB.

SinB = AC / 2 * R = 48/50 = 24/25.

Let’s define CosB.

Cos2B = 1 – Sin2B = 1 – 576/625 = 49/625.

CosB = 7/25.

Let’s apply the cosine theorem.

AC ^ 2 = AB ^ 2 + BC ^ 2 – 2 * AB * BC * CosB = 2 * AB ^ 2 – 2 * AB ^ 2 * CosB = 2 * AB ^ 2 * (1 – CosB).

2304 = 2 * AB ^ 2 * (1 – 7/25).

AB ^ 2 = 2304/2 * (18/25) = 1600.

AB = 40 cm.

Determine the area of the triangle ABC.

Savs = AB * BC * SinB / 2 = (40 * 40 * 24/25) / 2 = 768 cm2.

Then the radius of the inscribed circle is equal to: O1K = O1H = 2 * S / P = 2 * 768/128 = 12 cm.

In a rectangular triangle OCH: OH ^ 2 = OC ^ 2 – CH ^ 2 = 625 – 576 = 49. OH = 7 cm.

Then OO1 = O1H – OH = 12 – 7 = 5 cm.

Answer: Between the centers of the circles is 5 cm.

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