The height of the isosceles trapezoid is 9cm, and the middle line is 12cm. Find the diagonal of the trapezoid.

Let us prove that the middle line of the trapezoid is equal to the length of the DC segment.

The middle line of the trapezoid is equal to the half-sum of the base. KP = (BC + AD) / 2.

AD = AK + KH + DH.

Since the trapezoid is isosceles, then AK = DH, and since the VSNK is a rectangle, then KH = BC, then:

AD = 2 * DH + BC.

KP = (BC + 2 * DH + BC) / 2 = BC + DH = KD.

Then, in a right-angled triangle BDK, by the Pythagorean theorem, BD ^ 2 = BK ^ 2 + DK ^ 2 = 9 ^ 2 + 12 ^ 2 = 81 + 144 = 225.

ВD = 15 cm.

Answer: The diagonal of the trapezoid is 15 cm.