In an isosceles trapezoid ABCD with bases BC and AD, the side is a, the diagonal AC

In an isosceles trapezoid ABCD with bases BC and AD, the side is a, the diagonal AC is the bisector of the angle BAD, and the angle BAC is b. Find AD.

Since AC is the bisector of the angle BAD, the triangle ABC is isosceles, AB = BC = a cm.

Angle ABC = 2 * β0.

Let’s draw the height of the ВН.

In a right-angled triangle ABН, Cos (2 * β) = AH / AB.

AH = AB * Cos (2 * β) = a * Cos (2 * β) see.

In an isosceles trapezoid, the height drawn to the larger base is divided into two segments, the length of the smaller of which is equal to the half-difference of the lengths of the bases.

AH = (AD – BC) / 2.

AD = 2 * AH + BC = 2 * a * Cos (2 * β) + a = a * (1 + 2 * Cos (2 * β)).

Answer: The length of the base AD is equal to a * (1 + 2 * Cos (2 * β)) see.