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

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 lower the height from the top of the obtuse angle C to the larger base AD.

By the property of an isosceles trapezoid, the height lowered from the apex of an obtuse angle to the larger base divides it into two segments, the smaller of which is equal to the half-difference of the bases, and the larger half-sum of the bases.

DН = (АD – ВС) / 2 = (104 – 56) / 2 = 48/2 = 24 cm.

AH = (AD + BC) / 2 = (104 + 56) / 2 = 160/2 = 80 cm.

From the right-angled triangle СНD, according to the Pythagorean theorem, we define the leg СН, which is the height of the trapezoid.

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 determine the length of the hypotenuse AC.

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

AC = √6724 = 82 cm.

Answer: The diagonal of the trapezoid is 82 cm.