# In an isosceles trapezoid ABCD with a large base AD, the diagonal AC is the bisector of angle A

**In an isosceles trapezoid ABCD with a large base AD, the diagonal AC is the bisector of angle A and is perpendicular to the CD side. Find AD if the perimeter of the trapezoid is 25 cm.**

Since the trapezoid ABCD is isosceles, its angles at the base of AD are equal, BAD = CDA.

The diagonal AC, by condition, is the bisector of the angle BAD, therefore, BAC = CAD = CDA / 2.

Consider a right-angled triangle ACD, the sum of its angles is 180, CAD + 2 * CAD + 90 = 180.

3 * CAD = 180 – 90 = 90.

CAD = 30.

In a right-angled triangle, the leg opposite the angle 30 is half the length of the hypotenuse. HELL = 2 * CD.

Consider a triangle ABC, in which the angle BCA = CAD, and therefore equal to BAC, as lying crosswise at the intersection of two parallel lines of the secant AC. Then the triangle ABC is isosceles and AB = BA, but since the trapezoid is isosceles, then CD = AB = BA.

P = AB + BC + CD + AD = 5 * AB = 25.

AD = 25/5 = 5 cm.

AB = BC = CD = 5 cm.

AD = 5 * 2 = 10 cm.

Answer: AD = 10 cm.