# In trapezoid ABCD, the diagonal AC is perpendicular to the lateral side CD and is the bisector of angle A.

In trapezoid ABCD, the diagonal AC is perpendicular to the lateral side CD and is the bisector of angle A. Find the length AB if the perimeter of the trapezoid is 35 cm.Angle D = 60 Degrees

Consider triangle ACD.

According to the condition, the angle ADC is 60, the angle ACD = 90, since the diagonal AC is perpendicular to the side of CD, then the angle CAD = 180 – 90 – 60 = 30. Then the leg CD is equal to half of the hypotenuse AD. AD = 2 * CD.

Since AC, according to the condition, is the bisector of angle A, then the angle BAD = 2 * CAD = 2 * 30 = 60.

The angle BAD = CDA, and therefore the trapezoid is isosceles. AB = CD.

Consider a triangle ABC. Angle BAC = CAD, since AC is bisector. Angle BCA = CAD as cross-cutting angles at the intersection of parallel straight lines BC and AD secant AC, then the angle BAC = BCA, and therefore triangle ABC is isosceles. AB = BC.

Let’s designate the side AB through X, then AB = BC = CD = X, and AD = 2 * X.

P = AB + BC + CD + AD = X + X + X + 2 * X = 5 * X = 35.

X = 35/5 = 7 cm.

AB = 7 cm. 