The center of the circle circumscribed about the trapezoid lies on one of the bases of the trapezoid.

| September 21, 2021 | education

The center of the circle circumscribed about the trapezoid lies on one of the bases of the trapezoid. Find the area of a trapezoid if its lateral side is 4 cm, and one of its angles is 120 degrees.

Let’s draw the diagonal BD of the trapezoid. The resulting triangle ABD is rectangular, angle B = 90, as it rests on the diameter of the circle.

The sum of the angles at the lateral sides of the trapezoid is 180, then the angle BAD = 180 – 120 = 60, and the angle ADB = 180 – 90 – 60 = 30.

Leg AB of triangle ABD lies opposite angle 30, therefore, equal to half the length of the hypotenuse. Then AD = AB * 2 = 4 * 2 = 8 cm.

Let us omit from the top of the angle B of the trapezoid, the height BH. In the triangle ABН, the leg BН = AB * SinA = 4 * Sin60 = 4 * √3 / 2 = 2 * √3 cm.

The leg AH of the triangle ABН lies opposite the angle 30, therefore, AH = AB / 2 = 4/2 = 2 cm.

Since only an isosceles trapezoid can be inscribed in a circle, the base BC = (AD – 2 * AH) = 8 – 4 = 4 cm.

Determine the area of ​​the trapezoid.

S = (АD + ВС) * ВН / 2 = (8 + 4) * 2 * √3 / 2 = 12 * √3 cm2.

Answer: The area of ​​the trapezoid is 12 * √3 cm2.



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