The diagonals of the trapezoid are mutually perpendicular and equal to 12 and 18 cm. What is the area of the trapezoid?

Through the vertex C of the trapezoid, draw a segment CK parallel to the ВD diagonal.

In the quadrangle of the ВСКD, the opposite sides are parallel, then the ВСКD is a parallelogram.

BC = DC, СK = ВD.

Since the diagonals are perpendicular, and the CК is parallel to the ВD, the CК is perpendicular to the AC, and then the triangle ACK is rectangular.

Let’s build the height of the CH.

The area of the triangle Sask = AC * SC / 2 = 18 * 12/2 = 108 cm2.

Also Sask = AK * CH / 2.

AK = AD + DK = AD + BC.

Then Sask = (AD + BC) * CH / 2, which is equal to the area of the trapezoid.

Savsd = Sask = 108 cm2.

Answer: The area of the trapezoid is 108 cm2.