In trapezoid ABCD, diagonal BD is perpendicular to the side of AB, ADB = BDC = 30 degrees

| February 7, 2021 | education

In trapezoid ABCD, diagonal BD is perpendicular to the side of AB, ADB = BDC = 30 degrees Find the length AD if the perimeter of the trapezoid is 60 cm.

We denote the length of the lateral side AB through X.

Consider a right-angled triangle AВD, in which, by condition, the angle AВD = 90, and the angle ABD = 30, then the leg AB of a right-angled triangle is equal to half the length of its hypotenuse. AD = 2 * AB = 2 * X.

Determine the value of the angle ВAD. ВAD = 180 – ВDA – AВD = 180 – 30 – 90 = 60.

The SDA angle of the trapezoid is equal to ADВ + ВDС = 30 + 30 = 60.

Since the angles of ВAD and СDA are equal, the trapezoid of AВСD is isosceles, and therefore сD = AB = X cm.

Consider the ВСD triangle, in which the ВСD angle = 300, the СВD angle = ВDA = 30, as the angles lying crosswise at the intersection of parallel lines AD and BC of the secant ВD.

Then the angle СВDB =СDВ = 30, which means that the triangle of the ВСD is isosceles and BC = СD = X cm.

The perimeter of the trapezoid is: Ravsd = AB + BC + СD + AD = X + X + X + 2 * X = 60 cm.

5 * X = 60.

X = 60/5 = 12 cm.

Then AD = 2 * 12 = 24 cm.

Answer: AD length is 24 cm.



One of the components of a person's success in our time is receiving modern high-quality education, mastering the knowledge, skills and abilities necessary for life in society. A person today needs to study almost all his life, mastering everything new and new, acquiring the necessary professional qualities.