Find the height of an isosceles trapezoid if its bases are 12cm and 6cm, and the lateral

Find the height of an isosceles trapezoid if its bases are 12cm and 6cm, and the lateral side forms an angle of 150 ° with one of the bases.

The larger base of the trapezoid is equal to the sum of the lengths of the smaller base and the projections of the two lateral sides. Since this trapezoid is isosceles, the projections of its lateral sides onto the larger base are equal to each other and equal to half the difference in the lengths of the bases: (12 – 6) / 2 = 3 cm.

The sum of the angles of the trapezoid adjacent to the lateral side is 180 °. In this case, an obtuse angle lies between the lateral side and the smaller base, an acute one – between the lateral side and the larger base. Thus, the angle between the lateral side and the large base is:

180 ° – 150 ° = 30 °.

In a right-angled triangle formed by the lateral side, its projection and the height of the trapezoid, the projection of the lateral side is the leg adjacent to an angle of 30 °, the height is the leg opposite to this angle. The ratio of the opposite leg to the adjacent leg is the tangent of the angle. Means:

tg 30 ° = h / 3;

h = 3 * tan 30 ° = 3 * √3 / 3 = √3 ≈ 1.73 cm – the height of the trapezoid.