Find the perimeter of an isosceles trapezoid ABCD if you know that the measure of the angle is A = 30

Find the perimeter of an isosceles trapezoid ABCD if you know that the measure of the angle is A = 30, the smaller base is 2√3 and the height is 1 cm.

Let’s draw two heights of the trapezoid. Let’s get a projection of the upper base to the lower one.

Let’s denote one of the smaller segments of the lower (larger) base as x. Then this base itself is equal to 2√3 + 2x or 2 (x + √3).

We have an acute angle of 30 degrees, which means that the height of the trapezoid is equal to half of its lateral side. Those. side a is 2 * 1 = 2.

By the Pythagorean theorem, we find the side of the trapezoid.

a = x ^ 2 + 1 ^ 2 = x ^ 2 + 1.

Substitute the found value a into the equality.

2 = x ^ 2 + 1.

x ^ 2 = 1.

x = | 1 |, taking into account the conditions of the problem, x = 1.

Then the larger base of the trapezoid is 2 (1 + √3) = 2 + 2√3.

P = 2 * 2 + 2 + 2√3 + 2√3 = 6 + 4√3.

Answer: P = 6 + 4√3.