The diagonal AC of a rectangular trapezoid ABCD is perpendicular to the lateral side of CD

| May 1, 2021 | education

The diagonal AC of a rectangular trapezoid ABCD is perpendicular to the lateral side of CD and makes an angle of 60 ° with the base of AD. Find the area of the trapezoid if AD = 24cm.

Since the AC is perpendicular to the СD, the ACD triangle is rectangular, in which the ADC angle = 180 – 90 – 60 = 30. Then the AC leg lies opposite the angle 30 and is equal to half the length of the BP diagonal.

AC = AD / 2 = 24/2 = 12 cm.

In a right-angled triangle ABC, the angle BAC = BAD – CAD = 90 – 60 = 30. Then the leg BC lies opposite the angle 30 and is equal to half the length of the hypotenuse AC. BC = AC / 2 = 12/2 = 6 cm.

In a right-angled triangle ACН, the leg CH = AC * SinСAН = 12 * √3 / 2 = 6 * √3 cm.

Determine the area of the trapezoid.

Savsd = (ВС + АD) * СН / 2 = (6 + 24) * 6 * √3 / 2 = 90 * √3 cm2.

Answer: The area of the trapezoid is 90 * √3 cm2.



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