In an isosceles triangle ABC, sides AB = BC = 10, cos ABC = 7/25. Find the radius of the circle inscribed in this triangle ABC.

univerkov | August 15, 2021 | education

In the triangle ABC, by the cosine theorem, we define the length of the base of the AC.

AC ^ 2 = AB ^ 2 + BC ^ 2 – 2 * AB * BC * Cos ABC.

AC ^ 2 = 100 + 100 – 2 * 10 * 10 * 7/25 = 200 – 56 = 144.

AC = 12 cm.

Determine the area of the triangle ABC.

Sin2ABC = 1 – Cos2ABC = 1 – 49/625 = 576/625.

SinABC = 24/25.

Then Saavs = AB * BC * SinABC / 2 = 10 * 10 * 24/50 = 48 cm2.

Determine the radius of the inscribed circle.

R = 2 * Savs / P = 2 * 48 / (10 + 10 + 12) = 3 cm.

Answer: The radius of the inscribed circle is 3 cm.

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