A circle is inscribed in a triangle with a perimeter of 84. One of the tangency points divides the side of the triangle

| August 17, 2021 | education

A circle is inscribed in a triangle with a perimeter of 84. One of the tangency points divides the side of the triangle into segments with lengths 12 and 14. Find the area of the triangle.

By the property of a tangent drawn from one point, the lengths of the tangents are equal.

Then AM = AK = 14 cm, CH = SK = 12 cm.

The perimeter of the triangle is:

P = AM + BM + BH + CH + SK + AK = 84 cm.

84 = 14 + BM + BH + 12 + 12 + 14.

BM + BH = 84 – 52 = 32 cm.

Since ВМ = ВН as a segment of tangents, then ВМ = ВН = 32/2 = 16 cm.

AB = 14 + 16 = 30 cm.

AC = 14 + 12 = 26 cm.

BC = 12 + 16 = 28 cm.

Let us define the semiperimeter of the triangle. p = P / 2 = 84/2 = 42 cm.

By Heron’s theorem, we determine the area of a triangle.

Savs = Savs = √p * (p – AB) * (p – AC) * (p – BC) = √42 * 12 * 14 * 16 = √112896 = 336 cm2.

Answer: The area of the triangle is 336 cm2.



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