# The smaller side of an isosceles trapezoid is 15 cm and the height is 3√3 Find the area of a trapezoid

The smaller side of an isosceles trapezoid is 15 cm and the height is 3√3 Find the area of a trapezoid if one of its angles is 150 degrees.

1. AВСD – trapezoid; S = (a + b) * h / 2; 2. BO, CH – heights; BC = OH = 15 cm (by condition); BO = 3 * sqrt3; ∠ OBC = 90 degrees (OBCH is a rectangle), ∠ ABC + ∠ BAD = 180 degrees (the sum of the inner one-sided angles of the trapezoid) => ABC = 150 degrees (by condition), ∠ BAD = 180 – 150 = 30 degrees. 2. Consider a triangle ABO; ∠ BAD = ∠ BAO = 30 degrees => AB = 2 * (3 * √3) = 6 * √3 cm (a leg lying opposite an angle of 30 degrees is equal to half the hypotenuse); AB ^ 2 = AO ^ 2 + BO ^ 2; AO ^ 2 = (6 * √3) ^ 2 – (3 * √3) ^ 2 = 81; AO = 9 cm; 3. AD = AO + OH + HD; AO = HD = 9 cm; AD = 9 + 9 + 15 = 33 cm; 4.S = (15 + 33) * 3 * √3 / 2 = 72 * √3 cm2
Answer: 72 * √3 cm sq.

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