The bases of the isosceles trapezoid are equal to a and b the lateral side is equal to c. Find the diagonal of the trapezoid.

From the top B we construct the height BH to the base of AD. Since the trapezoid ABCD is isosceles, its height BH divides the larger base into two segments, the length of the larger of which is equal to half the sum of the lengths of the bases of the trapezoid.

DН = (ВС + АD) / 2 = (a + b) / 2 cm.

AH = (AD – BC) / 2 = (b – a) / 2 cm.

By the Pythagorean theorem, from the triangle ABN, BH ^ 2 = AB ^ 2 – AH ^ 2 = c ^ 2 – (b – a) ^ 2/4

By the Pythagorean theorem, from the triangle BDH, BD ^ 2 = BH ^ 2 + DH ^ 2 = c ^ 2 – (b – a) ^ 2/4 + (a + b) ^ 2/4 = C ^ 2 + (( b – a) ^ 2 + (a + b) ^ 2) / 4 = c ^ 2 + 4 * a * b / 4 = c ^ 2 + a * b.

BD = √ (c ^ 2 + a * b).

Answer: The diagonal of the trapezoid is √ (c ^ 2 + a * b)