Find the area of a trapezoid with bases 13 cm and 7 cm in diagonals 16 cm and 12 cm.

Draw a straight line CK through the vertex C of the trapezoid, parallel to the diagonal BD.

The quadrangle ВСКD is a parallelogram, since its opposite sides are parallel, then DК = ВС = 7 cm, СК = ВD = 12 cm.

AK = AD + DK = 13 + 7 = 20 cm.

By Heron’s theorem, we determine the area of the triangle ACK.

The half-perimeter of the triangle is: p = (16 + 12 + 20) / 2 = 24 cm.

Then Sask = 24 * (24 – 20) * (24 – 16) * (24 – 12) = 9216 = 96 cm2.

Let’s build the height of the CH.

The area of the triangle ACK is equal to: Sask = AK * CH / 2 = (AD + DC) * CH / 2 = (AD + BC) * CH / 2, which is the area of the trapezoid.

Savsd = Sask = 96 cm2.

Answer: The area of the trapezoid is 96 cm2.