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.

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.