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

| July 26, 2021 | education

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.



One of the components of a person's success in our time is receiving modern high-quality education, mastering the knowledge, skills and abilities necessary for life in society. A person today needs to study almost all his life, mastering everything new and new, acquiring the necessary professional qualities.