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

| April 10, 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, angle D = 60 degrees.

In triangle АСD, according to the condition, the angle АDC = 60, and the angle АСD = 90, then the angle АСD = 180 – 90 – 60 = 30.

Since AC is the bisector of the angle BAD, the angle BAD = BAC + CAD = 30 + 30 = 60, therefore, the trapezoid is isosceles and AB = CD.

Let the length AB = X cm.

In triangle АСD, by condition the angle АDC = 60, and АС is perpendicular to СD, then the angle CAD = 180 – 90 – 60 = 30.

Leg CD = AB = X cm, and lies against an angle of 30, then the hypotenuse AD is equal to the length of two legs CD.

AD = 2 * CD = 2 * X.

Then the perimeter of the trapezoid is:

P = AB + BC + CD + AD = X + X + X + 2 * X = 5 * X = 35 cm.

X = 35/5 = 7 cm.

AB = BC = CD = 5 cm.

Answer: Length AB = 5 cm.



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