In trapezoid ABCD, the diagonal AC is perpendicular to the side of CD and bisects angle A, angle D = 60 degrees.

| February 4, 2021 | education

In trapezoid ABCD, the diagonal AC is perpendicular to the side of CD and bisects angle A, angle D = 60 degrees. Find the middle line of this trapezoid if its perimeter is 20 cm.

The AСD triangle, by condition, is rectangular. Then the angle DAС = 180 – 90 – 60 = 30.

Since the angle BAC = DAС, then the angle BAD = 2 * DAС = 2 * 30 = 60, then the angles at the base of AD are equal, then the trapezoid is isosceles and AB = СD.

The diagonal AC is the bisector of the angle BAC, then it cuts off the isosceles triangle ABC, in which AB = BC.

In a right-angled triangle of AСD, the leg of the СD lies against an angle of 300, then СD = AD / 2.

AD = 2 * СD.

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

The perimeter of the trapezoid is then: Rasvd = X + X + X + 2 * X = 20 cm.

5 * X = 20 cm.

X = AB = BC = СD = 20/5 = 4 cm. AD ​​= 2 * 4 = 8 cm.

Determine the length of the midline of the trapezoid.

KM = (BC + AD) / 2 = (4 + 8) / 2 = 6 cm.

Answer: The length of the middle line is 6 cm.



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