The radius of a circle circumscribed about an equilateral triangle is 8 cm. Find the perimeter

| February 14, 2021 | education

The radius of a circle circumscribed about an equilateral triangle is 8 cm. Find the perimeter of the triangle and the radius of the inscribed circle.

The radius of the circumscribed circle in a triangle is the intersection point of the median perpendiculars, and since the ABC triangle is equilateral, then its heights, medians and bisectors of the triangle.

The medians of the triangle, at the point of their intersection, are divided by a ratio of 2/1 starting from the apex of the triangle. Then OB = R = 2 * OH.

OH = r = R / 2 = 8/2 = 4 cm.

Then ВН = OB + OH = 8 + 4 = 12 cm.

The height of an equilateral triangle is: BH = AC * √3 / 2, then

AC = 2 * ВН / √3 = 2 * 12 / √3 = 8 * √3 cm.

Then the perimeter of the triangle ABC is: P = 3 * AC = 3 * 8 * √3 = 24 * √3 cm.

Answer: The perimeter of the triangle is 24 * √3 cm, the radius of the inscribed circle is 4 cm.



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