Isosceles trapezoid, larger base = 4 cm, diagonal = 3 cm, side = 2 cm. Find the area.

univerkov | January 4, 2021 | education

Consider an AВD triangle for which the lengths of all sides are known. Let us define its area by Heron’s theorem.
Determine the semi-perimeter of the AED triangle. P = (2 + 3 + 4) / 2 = 9/2 = 4.5 cm.
Then Savs = √r * (p – AB) * (p – ВD) * (p – AD) = √4.5 * 2.5 * 1.5 * 0.5 = √ (135/16) = √135 / 4.
Also Savd = AD * ВН / 2.
ВН = 2 * Savd / BP = (2 * √135 / 4) / 4 = √135 / 8 = 1.45 cm.
In a right-angled triangle ABН, according to the Pythagorean theorem, AH2 = AB2 – BH2 = 4 – 2.1 = 1.9.
AH = 1.38 cm.
Since the trapezoid is isosceles, then AH = (AD – BC) / 2.
BC = AD – 2 * AH = 4 – 2.76 = 1.24 cm.
Then Savsd = (ВС + AD) * ВН / 2 = 3.8 cm2.
Answer: The area of the trapezoid is 3.8 cm2.

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