In equilateral trapezoids, the diagonal is the bisector of an acute angle and forms an angle of 30

In equilateral trapezoids, the diagonal is the bisector of an acute angle and forms an angle of 30 with a large base. Find the perimeter if a large base is given 4.

Since, about the condition, AC is the bisector of angle A, then the angle DAB = 2 * DAC = 2 * 30 = 60.

Then the angle ADC = 60, since the trapezoid is isosceles.

In triangle CD, angle ACD = 180 – 60 – 30 = 90, therefore, triangle ACD is rectangular, in which the leg CD lies opposite angle 30. Then CD = AD / 2 = 4/2 = 2 cm.

Then AB = CD = 2 cm.

Since the diagonal AC and the bisector of angle A, then BC = AB = 2 cm.

Determine the perimeter of the trapezoid. Ravsd = AB + BC + CD + AD = 2 + 2 + 2 + 4 = 10 cm.

Answer: The perimeter of the trapezoid is 10 cm.