An isosceles triangle is inscribed in a circle with a radius of 8. the angle at the base of the triangle

| May 16, 2021 | education

An isosceles triangle is inscribed in a circle with a radius of 8. the angle at the base of the triangle is 75. find the area of the triangle.

In an isosceles triangle ABC, we determine the value of the angle ABC.

Angle ABC = (180 – 75 – 75) = 30.

The inscribed angle ABC rests on the arc AC, then the arc AC = 2 * 30 = 60. Then the central angle AOC = 60, and the triangle AOC will be equilateral, AC = OC = AC = 8 cm.

In an equilateral triangle ABC, we apply the cosine theorem.

Let AB = BC = X cm.

Then:

AC ^ 2 = X ^ 2 + X ^ 2 – 2 * X * X * Cos30.

64 = 2 * X ^ 2 – X ^ 2 * √3.

X ^ 2 * (2 – √3) = 64.

X ^ 2 = 64 / (2 – √3) = AB * BC.

Then Saavs = AB * BC * Sin30 / 2 = (64 / (2 – √3) * (1/2) / 2 = 16 / (2 – √3) cm2.

Answer: The area of the triangle is 16 / (2 – √3) cm2.



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