Find the area of a rectangle if its diagonal is 12√3 cm and makes an angle of 60 ° with its side.

Consider one of two identical triangles into which the diagonal divides this rectangle.

Such a triangle is rectangular, the legs of which are the sides of the rectangle, and the hypotenuse is the diagonal of the rectangle.

According to the condition of the problem, the length of the hypotenuse is 12√3 cm, and the angle between the hypotenuse and one of the legs is 60 °.

Applying the formulas of a right-angled triangle, we find the lengths of the legs:

12√3 * sin (60 °) = 12√3 * 1/2 = 6√3 cm;

12√3 * cos (60 °) = 12√3 * √3 / 2 = 6 * 3 = 18 cm.

Knowing the lengths of the sides of the rectangle, we find its area:

6√3 * 18 = 108√3 cm ^ 2.

Answer: 108√3 cm ^ 2.