In a trapezoid in ABCD with a large base AD, the diagonal AC is perpendicular to the side of the CD

| September 7, 2021 | education

In a trapezoid in ABCD with a large base AD, the diagonal AC is perpendicular to the side of the CD, angle BAC = angle CAD. Find AD if the perimeter of the trapezoid is 20cm, which is, and the Angle D = 60 °.

Since AC is perpendicular to CD, the ACD triangle is rectangular, then the angle DAC = 180 – 90 – 60 = 30. Since, by condition, the angle BAC = CAD, then the angle BAD = 2 * 30 = 60, then the trapezoid ABCD is isosceles. AB = CD, and since AC is a bisector, it cuts off the isosceles triangle at the lateral side AB, then AB = BC = CD.

In a right-angled triangle ACD, the leg CD lies opposite the angle 30, then AD = 2 * CD.

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

Then the perimeter of the trapezoid is: Ravsd = X + X + X + 2 * X = 20 cm.

5 * X = 20.

X = AB = BC = CD = 20/5 = 4 cm.

AD = 2 * 4 = 8 cm.

Answer: The length of the AD base is 8 cm.



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