The diagonal of the rectangle is 10√2 cm and the angle between the diagonals is 45 degrees, find the area of the rectangle.

First way.

The area of a rectangle is half the square of the diagonal times the sine of the angle between the diagonals.

S = AD ^ 2 * Sin450 / 2 = ((10 * √2) ^ 2 * √2 / 2) / 2 = 200 * √2 / 4 = 50 * √2 cm2.

Second way.

The diagonals of a rectangle divide it into four isosceles triangles, whose equal sides are equal to half the length of the diagonals. ОА = ОВ = ОВ = ОD = 5 * √2 cm.

Soav = Scod = AO ^ 2 * Sin450 / 2 = ((5 * √2) ^ 2 * √2 / 2) / 2 = 12.5 * 5 * √2 cm2.

Swos = Saod = AO ^ 2 * Sin1350 / 2 = ((5 * √2) ^ 2 * √2 / 2) / 2 = 12.5 * 5 * √2 cm2.

Savsd = 2 * Sоav + 2 * Svos = 50 * √2 cm2.

Answer: The area of the rectangle is 50 * √2 cm2.