In an isosceles trapezoid, the middle line is 12 cm, the height is 5 cm. Find the diagonal of the trapezoid.

ABCD – isosceles trapezoid: AB = CD – lateral sides, AD = a and BC = b – larger and smaller bases, respectively, BH = 5 cm – height.
1. Height BH divides the base AD into segments AH = (AD – BC) / 2 and DH = (AD + BC) / 2.
Since the middle line of the trapezoid is equal to the half-sum of the bases (m = (a + b) / 2 = (AD + BC) / 2), then:
m = DH = (AD + BC) / 2 = 12.
2. Consider a right-angled triangle BHD: angle BHD = 90 degrees, BD – hypotenuse (trapezoidal diagonal), BH = 5 cm and DH = 12 cm – legs.
By the Pythagorean theorem:
BD = √ (BH ^ 2 + DH ^ 2) = √ (5 ^ 2 + 12 ^ 2) = √ (25 + 144) = √169 = 13 (cm).
Since the diagonals of an isosceles trapezoid are equal, then BD = AC = d = 13 cm.
Answer: d = 13 cm.