The bases of an isosceles trapezoid are 12 and 42, the side is 39. Find the length of the diagonal of the trapezoid.

Let an isosceles trapezoid ABCD with bases AD = 42 and BC = 12, lateral sides AB = CD = 39 cm be given. = АD – МD = 42 – 15 = 27. Consider Δ ABK formed by the height BK. In it, according to the Pythagorean theorem, the following relation is fulfilled: AB ^ 2 = AK ^ 2 + BK ^ 2 or 39 ^ 2 = 15 ^ 2 + BK ^ 2, then BK = CM = 36. To find the diagonal AC, we apply the Pythagorean theorem to Δ AFM: AC ^ 2 = AM ^ 2 + CM ^ 2 or AC ^ 2 = 27 ^ 2 + 36 ^ 2, then AC = 45.
Answer: the length of the diagonal of the trapezoid AC = 45.