The bases of the trapezoid are 3 cm and 11 cm, and the diagonals are 13 cm and 15 cm. Find the area of the trapezoid

| February 1, 2021 | education

From the vertex “C” of the trapezoid, draw a straight line CК parallel to the ВD, and extend the base of the ABP until it intersects with the CК. The resulting quadrangle of the ВСКD is a parallelogram, and therefore BC = DC = 3 cm, ВD = SC = 15 cm.

The area of ​​the trapezoid is equal to the sum of the areas of the triangles ABC and AСD.

The area of ​​the triangle ABC is equal to: Savs = BC * НK / 2.

Consider a triangle СDK, the area of ​​which is equal to: Sсдк = DК * НК / 2. Since DК = ВС, then Sас = Sсдк.

Then the area of ​​the trapezoid is equal to: Savsd = Sasd + Ssdk = Sask.

The area of ​​the triangle AСK is determined by Heron’s theorem.

S = √p * (p – AC) * (p – CK) * (p – AK), where p is the semiperimeter of the triangle.

p = (AC + СK + AK) / 2 = (13 + 15 + 14) / 2 = 21 cm.

S = √21 * (21 – 13) * (21 – 15) * (21 – 14) = √7056 = 84 cm2.

Savsd = 84 cm2.

Answer: The area of ​​the trapezoid is 84 cm2.



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