Regular triangle and quadrangle are inscribed in a circle. The perimeter of a triangle is 6√6.

Regular triangle and quadrangle are inscribed in a circle. The perimeter of a triangle is 6√6. Find the perimeter of the quadrilateral.

Since triangle ABC is regular, we determine its side length through the perimeter.

AB = BC = AC = P / 3 = 6 * √6 / 3 = 2 * √6.

Let us determine the radius of the circle described around the triangle ABC.

R = ОА = AB / √3 = 2 * √6 / √3 = 2 * √2 * √3 / √3 = 2 * √2 cm.

A regular quadrilateral AKMD is a square, then its diagonals are equal and intersect at right angles, then triangle AOD is rectangular, and OD = OA = 2 * √2 cm.

Then, by the Pythagorean theorem, AD ^ 2 = ОА ^ 2 + ОD ^ 2 = 8 + 8 = 16.

AD = 4 cm.

Since AK = KM = MD = AD, then Pkv = 4 * AD = 4 * 4 = 16 cm.

Answer: The perimeter of the quadrangle is 16 cm.