A perpendicular MA is drawn through the vertex A to the plane of the square ABCD

A perpendicular MA is drawn through the vertex A to the plane of the square ABCD, the angle between the straight line MC and the plane of the square is 45 degrees, and MA is 4√2 cm. Find the area of the square.

Since AM is perpendicular to the plane of the square, then the MAC triangle is rectangular, then the angle AFM = 180 – 90 – 45 = 45, therefore the AFM triangle is also isosceles AM = AC = 4 * √2 cm.

Since ABCD is a square, then AB = BC = CD = AD. In a right-angled triangle ABC, according to the Pythagorean theorem, we determine the lengths of the legs AB and BC.

AB ^ 2 + BC ^ 2 = AC ^ 2.

2 * AB ^ 2 = (4 * √2) ^ 2 = 32.

AB ^ 2 = 16.

AB = BC = CD = AD = 4 cm.

The area of the square is equal to AB ^ 2 = 4 ^ 2 = 16 cm2.

Answer: The area of the square is 16 cm2.