The bases of the trapezoid are 10 and 5, and the diagonals are 9 and 12. Find the area of the trapezoid

univerkov | January 26, 2021 | education

Let’s carry out an additional construction. From the vertex C of the trapezium, draw a segment parallel to the ВD diagonal, until it intersects with the ABP at point K. The figure of the ВСKD is a parallelogram, then DK = ВС = 5 cm, СK = ВD = 9 cm.

AK = AD + DC = 10 + 5 = 15 cm.

The area of the trapezoid is equal to: Savsd = (BC + AD ) * CH / 2.

Triangle area ACK = AK * CH / 2 = (AD + DK) * CH / 2 = (BC + AD ) * CH / 2.

The area of the trapezoid is equal to the area of the triangle ACK, which we define by Heron’s theorem.

Sawk = √p * (p – a) * (p – b) * (p – c), where p is the half perimeter of the triangle, a, b, c are the lengths of the sides of the triangle.

P = (12 + 9 + 15) / 2 = 18 cm.

Savk = √18 * (18 – 15) * (18 – 12) * (18 – 9) = √18 * 3 * 6 * 9 = √2916 = 54 cm2.

Savsd = 54 cm2.

Answer: The area of the trapezoid is 54 cm2.

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