The bases of the isosceles trapezoid 15 and 27 and its area 168 find the side.

1. When calculating we use the formula: the area of ​​the trapezoid is equal to the product of the half-sum of the bases and the height. Let us denote the given trapezoid by the letters ABCD.

2. From the vertices B and C we omit the perpendiculars BO and CE, we got two right-angled triangles ABO and CED, in which AO = ED = (27 – 15): 2 = 6, because the trapezoid is isosceles.

By the condition of the problem, it is known that the area is equal to 168, that is

(BC + AD) * BO: 2 = 168, whence BO = 168 * 2: (BC + AD) = 168 * 2: 42 = 4 * 2 = 8.

Then we calculate the side AB by the Pythagorean theorem

AB² = AO² + BO², that is, AB = √6² + 8² = √100 = 10.

Answer: The side is 10.