In an isosceles trapezoid ABCD, the base AD = 8 cm, the diagonal BD is perpendicular

| May 15, 2021 | education

In an isosceles trapezoid ABCD, the base AD = 8 cm, the diagonal BD is perpendicular to the lateral side AB and the angle at the base AB = 60 degrees, find the area of the trapezoid.

Since the diagonal BD is perpendicular to the lateral side AB, the triangle ABD is rectangular, and the angle ADB = 180 – 90 – 60 = 30.

The AB leg lies opposite the angle 30, then its length will be: AB = AD / 2 = 8/2 = 4 cm.

Let’s draw the height of the HВ trapezoid. In a right-angled triangle ABН, the angle ABН = 180 – 90 – 60 = 30, then the length of the leg AH = AB / 2 = 4/2 = 2 cm.

Determine the length of the HВ height.

BH ^ 2 = AB ^ 2 – AH ^ 2 = 16 – 4 = 12.

BH = 2 * √3 cm.

The height of an isosceles trapezoid divides the base into two segments, the length of the smaller of which is equal to the half-difference of the bases. AH = (AD – BC) / 2.

2 * AH = AD – BC.

BC = AD – 2 * AH = 8 – 2 * 2 = 4 cm.

Determine the area of ​​the trapezoid.

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

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



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