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.

| January 15, 2021 | education

Let’s make additional constructions. Let’s draw a segment CE parallel to the diagonal ВD and connect point D with point E. Since the resulting figure is a parallelogram, then ВC = DE = 15 cm, and ВD = CE = 12 cm.
The area of ​​a trapezoid is equal to the sum of the areas of two triangles. Str = Savs + Sasd.
The area of ​​the triangle ABC can be determined by the formula: Sавс = ВС * НD / 2.
Determine the area of ​​the triangle СDE. Ssde = DE * НD / 2.
Since DE = BC, then Savs = Ssde.
Then Str = Sacd + Ssde = Sace.
The area of ​​the ACE triangle is determined by Heron’s theorem.
Sace = √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 = (AC + CE + AE) / 2 = (13 + 15 + 14) / 2 = 21 cm.
Sace = √21 * (21 – 13) * (21 – 15) * (21 – 14) = Sace = √21 * 8 * 6 * 7 = √7056 = 84 cm2.
Sta = Sace = 84 cm2.
Answer: The area of ​​the trapezoid is 84 cm2.



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