The bases of the isosceles trapezoid are 56 and 104, the lateral side is 30. Find the length of the diagonal of the trapezoid.

Let us draw the height CH from the top of the obtuse angle C.

In a rectangular trapezoid, the height drawn from the top of an obtuse angle divides the larger base into two segments, the smaller of which is equal to the half-difference of the bases of the trapezoid, and the larger one – half the sum of the bases.

DН = (АD – ВС) / 2 = (104 – 56) / 2 = 24 cm.

AH = (AD + BC) / 2 = (104 + 56) / 2 = 80 cm.

From the right-angled triangle СDН, according to the Pythagorean theorem, we define the leg СН.

CH ^ 2 = CD ^ 2 – DH ^ 2 = 30 ^ 2 – 24 ^ 2 = 900 – 576 = 324.

CH = √324 = 18 cm.

From the right-angled triangle ACH, according to the Pythagorean theorem, we define the hypotenuse AC.

AC ^ 2 = AH ^ 2 + CH ^ 2 = 80 ^ 2 + 18 ^ 2 = 6400 + 324 = 6724.

AC = √6724 = 82 cm.

Answer: The length of the diagonal of the trapezoid is 82 cm.