The obtuse angle of the parallelogram is 120 °. The height of the parallelogram drawn from this angle

The obtuse angle of the parallelogram is 120 °. The height of the parallelogram drawn from this angle is 6√3 cm and divides the side of the parallelogram in the ratio 1: 2, if counted from the top of the acute angle. Find the perimeter of the parallelogram.

The degree measure of the parallelogram angle adjacent to the 120 ° angle is 180 ° – 120 ° = 60 °.

In a right-angled triangle formed by the height drawn and the vertex of the obtuse angle of the parallelogram, by the segment connecting the apex of the acute angle of the parallelogram with the point of intersection of the height and side of the parallelogram and the side of the parallelogram connecting adjacent acute and obtuse angles, the ratio of the height to the segment connecting the vertex of the acute angle of the parallelogram and the point the intersection of the height and side of the parallelogram is tg 60 °, and the ratio of the height and side of the parallelogram is sin 60 °.

Since the height divides the side of the parallelogram in a ratio of 1: 2, the length of this side will be equal to three times the ratio of the height to tg 60 °:

3 * 6√3 / tg 60 ° = 3 * 6√3 / √3 = 18 (cm).

The second side of the parallelogram can be found as the ratio of the height to sin 60 °:

6√3 / sin 60 ° = 6√3 / (√3 / 2) = 12 (cm).

Find the perimeter of the parallelogram:

(18 + 12) * 2 = 60 (cm).

Answer: 60 cm.