The smaller side of the rectangle is 27, the diagonals intersect at an angle of 60 degrees. Find the diagonals of the rectangle.

The AOB triangle is isosceles and the angle at the apex of the isosceles triangle is 60 degrees, hence the angles at the base OAB = OBA = 60 degrees. We draw the height OK, which is the median for an isosceles triangle, it divides the side AB in half, which means AK = 1 / 2AB = 1/2 * 27 = 13.5. In the OKA triangle, by definition, cosOAK = AK: OA, which means OA = AK: cosOAK = 13.5: cos60 = 13.5: 1/2 = 27. We have, the diagonal AC = 2OA = 2 * 27 = 54., Hence BD = AC = 54, the diagonals of the rectangle are equal.