The smaller diagonal of a rectangular trapezoid is perpendicular to the side, the acute angle of the trapezoid is 45 degrees

The smaller diagonal of a rectangular trapezoid is perpendicular to the side, the acute angle of the trapezoid is 45 degrees, the larger base of the trapezoid is 8 cm. Find the area of the trapezoid.

The ACD triangle is rectangular and isosceles, since, by condition, AC is perpendicular to CD, and the angle ADC = 45.

Let’s build the height of CH, which is also the median of the triangle ACD, then AH = DH = AD / 2 = 8/2 = 4 cm.

The triangle CDH is rectangular and isosceles, then CH = DH = 4 cm.

Quadrangle ABCН is a rectangle, then BC = AH = 4 cm.

Determine the area of the trapezoid.

Savsd = (ВС + АD) * СН / 2 = (4 + 8) * 4/2 = 24 cm2.

Answer: The area of the trapezoid is 24 cm2.