# In an isosceles trapezoid, the side is 6 cm, the smaller side is 4 cm, and one of the corners is 120

In an isosceles trapezoid, the side is 6 cm, the smaller side is 4 cm, and one of the corners is 120 degrees. Find: area ABCD.

Consider a trapezoid ABCD – it is isosceles. the sides are equal, AB = CD = 6 cm, the smaller base CD = 4 cm, the angle B = 120 degrees (as the angle with the smaller base).
The sum of the angles adjacent to the side is 180 degrees, hence:
<A = 180- <B = 180-120 = 60
The angles at one base of an isosceles trapezoid are equal:
<A = <D, <B = <C
Let’s build the heights of the trapezoid ВK and CN, as seen AD = AK + KN + ND
Consider triangles ABK and CND, ВK = CN, AB = CD, <N = <K, <D = <A, triangles are equal in two sides and three angles, so AK = ND
Triangle ABK is rectangular.
cosA = AK / AB;
AK = cosA * AB = (1/2) * 6 = 3 cm.
sinA = BK / AB;
BK = sinA * AB = (√3 / 2) * 6 = 3√3 cm. (BK = h)
Back to the trapezoid
AD = AK + KN + ND = 3 + 4 + 3 = 10 cm.
S = ((BC + AD) / 2) * h = ((4 + 10) / 2) * 3√3 = 21√3 cm ^ 2
Answer: S = 21√3 cm ^ 2

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