The acute angle A of a rectangular trapezoid ABCD is 30 degrees. The sum of the lengths of its lateral sides

| May 25, 2021 | education

The acute angle A of a rectangular trapezoid ABCD is 30 degrees. The sum of the lengths of its lateral sides is 12√3, the smaller base BC is 8 cm. Calculate: a) the area of the trapezoid. b) the distance from vertex B to the diagonal.

Let’s build the height of the CH.

CH = AB since ABCН is a rectangle.

By condition, AB + CD = 12 * √3 cm, then CH + CD = 12 * √3 cm.

The leg CH of a right-angled triangle СDН lies opposite an angle of 30, then CD = 2 * CH.

CH = 12 * √3 – CD.

CD = 2 * (12 * √3 – CD).

СD = 8 * √3 cm.

CH = 4 * √3 cm.

Then DH = CD * Cos30 = 8 * √3 * √3 / 2 = 12 cm.

AD = BC + DH = 20 cm.

Determine the area of ​​the trapezoid

Savsd = (ВС + АD) * СН / 2 = (8 + 20) * 4 * √3 / 2 = 56 * √3 cm2.

Determine the area of ​​the triangle ABC. Savs = AB * BC / 2 = 4 * √3 * 8/2 = 16 * √3 cm2.

AC ^ 2 = AB ^ 2 + BC ^ 2 = 48 + 64 = 112.

AC = 4 * √7 cm.

Savs = AC * ВK / 2.

ВK = 2 * Savs / AC = 2 * 16 * √3 / 4 * √7 = 8 * √ (3/7) = 8 * √21 / 7 cm.

Answer: The area of ​​the trapezoid is 56 * √3 cm2, from point B to the diagonal 8 * √21 / 7 cm.



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