Dropping a perpendicular from the vertex of an equilateral triangle, with side length x, to the opposite side gives you two right triangles, each having hypotenuse of length x and one leg of length x/2. By the Pythagorean theorem, the other leg, the altitude of the equilateral triangle, is . Taking the distance from the vertex to the center of the circle, a radius of the circle and a diagonal of the square, to be y, the distance from the center of the circle to the opposite side is and now we have a right triangle with one leg of length , one leg of length x/2, and hypotenuse of length y. Set up the Pythagorean theorem for that triangle, solve for y. Now use the fact that is a square has diagonal of length y, then its side has length to find the area of the square.

Unfortunately, what you "understand" is not true. The angles in an equilateral triangle are 60 degrees. The legs of a right triangle with one angle 60 degrees and hypotenuse of length x/2 are x/4 and . The sides of the rectangle formed are x/2 and .my other question is similar

in an equilateral triangle ABC of sides 2x, a rectangle is constructed with the midpoints AB and AC as two vertices and the other two lying on bc. I understand one side is x but how do you find the breadth

thnx in advance