The base of a rectangular trapezoid is 18 cm and 12 cm, and the diagonal is the bisector

The base of a rectangular trapezoid is 18 cm and 12 cm, and the diagonal is the bisector of an acute angle. Find the diagonal.

Since BD, by condition, is the bisector of the ADC angle, it cuts off the isosceles triangle BCD. СD = ВС = 12 cm.

Let’s build the height of the CH. Quadrangle ABCH is a rectangle, then AH = BC = 12 cm.

DН = АD – АН = 18 – 12 = 6 cm.

In a right-angled triangle СDН, the leg DН = СD / 2, then the angle DСН = 30.

Then the angle ВСD = ВСН + DСН = 90 + 30 = 120.

In the triangle ВСD, by the cosine theorem, we define the length of the side ВD.

ВD ^ 2 = ВС ^ 2 + СD ^ 2 – 2 * ВС * СD * Cos120 = 144 + 144 – 2 * 12 * 12 * (-1/2) = 3 * 144.

ВD = 12 * √3 cm.

Answer: The length of the diagonal is 12 * √3 cm.