Find the area of a trapezoid whose bases are 6 and 12 cm, and the side 8 cm long forms an angle of 120 degrees with the smaller base.

Let the trapezoid be ABCD, with bases BC (6 cm) and AD (12 cm). The smaller base is the top. Let’s draw the height CH to the base of AD. Because angle С = 120 => angle D = 60 (by the trapezoid property). Consider the triangle CHD. It is rectangular (angle CHD = 90), so the angle is HCD = 30 (180-60-90) => HD = 0.5CD = 4 cm (by the property of a right-angled triangle). By the Pythagorean theorem: CH = √ (CD ^ 2-CH ^ 2) = √ (64-16) = √48 = 4√3 S = 0.5 * CH * (BC + AD) = 36√3
Answer: 36√3