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

| September 8, 2021 | education

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

Consider a right-angled triangle АСD, which, by condition, has an angle АСD = 90, and an angle АDC = 60, then the angle CAD = 180 – 90 – 60 = 30.

The leg CD of the triangle lies opposite the angle 30, then it is equal to half the length of the hypotenuse AD.

CD = AD / 2, then AD = 2 * CD.

Since, by condition, AC is the bisector of the angle, then the angle BAC = CAD = 30.

Angle ACB and angle CAD cross-lying angles at the intersection of parallel AD and BC secant AC, therefore they are equal.

ACB = CAD = BAC = 30, therefore, triangle ABC is isosceles and AB = BC.

Angle BAD = ABC + CAD = 30 + 30 = 60 = ADC, then the angles at the base of the trapezoid are equal, which means that the trapezoid is isosceles and AB = CD.

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

The perimeter of the trapezoid is: P = AB + BC + CD + AD = X + X + X + 2 * X = 35.

5 * X = 35.

X = 35/5 = 7 cm.

AB = 7 cm.

Answer: The length of the side AB = 7 cm.



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