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.
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