In trapezium ABCD, diagonal BD is perpendicular to the lateral side AB and is the bisector of angle D.

| April 11, 2021 | education

In trapezium ABCD, diagonal BD is perpendicular to the lateral side AB and is the bisector of angle D. The perimeter of trapezoid ABCD is 20 cm, angle A is 60 °. Find the length AD.

Consider a right-angled triangle ABD, in which the angle B is straight, and the angle A = 60, by condition. Then the angle ADB = 180 – 90 – 60 = 30.

Since, by condition, BD is the bisector of angle D, then the angle BDC = ADB = 30, then the angle ADC = ADB + BDS = 30 + 30 = 60.

Since the angles at the base of AD are equal, the trapezoid ABCD is isosceles and AB = CD.

In the triangle ВСD, the angle DBC = ADB = 30 as criss-crossing angles at the intersection of parallel straight lines AD and BC of the secant BD, then the angle DBC = BDC = 30, and the triangle BDC is isosceles and BC = CD.

In a right-angled triangle ABD, leg AB lies opposite angle 30, and therefore is equal to half the length of the hypotenuse AD, then AD = 2 * AB.

Let the length AB = X cm, then AB = BC = CD = X cm, and AD = 2 * X.

The perimeter of the trapezoid is: P = AB + BC + SD + AD = X + X + X + 2 * X = 20.

5 * X = 20.

X = 20/5 = 4 cm.

AB = 4 cm, then AD = 2 * 4 = 8 cm.

Answer: AD = 8 cm.



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