The bases of an isosceles trapezoid are 16 and 96, the side is 58. Find the length of the diagonal of the trapezoid.

Let a be the smaller base of the trapezoid, b the larger base, c and d the sides, d1 and d2 the diagonals of the trapezoid.
It is known from the properties of the trapezoid that the diagonals are related to the sides of the trapezoid by the ratio:
d1 ^ 2 + d2 ^ 2 = 2ab + c ^ 2 + d ^ 2.
Since the trapezoid given by the condition is isosceles, the lengths of its diagonals are equal (from the properties of an isosceles trapezoid), so you can make the replacement: d2 denote as d1. It is also known that the sides of an isosceles trapezoid are equal, so d can be replaced with c. Thus, the expression will turn out:
d1 ^ 2 + d1 ^ 2 = 2ab + c ^ 2 + c ^ 2.
Here are similar ones:
2 * d1 ^ 2 = 2ab + 2 * c ^ 2.
Let’s substitute the data we know into the expression:
2 * d1 ^ 2 = 2 * 16 * 96 + 2 * (58) ^ 2.
Let’s solve the resulting equation and find the length of the diagonal d1:
2 * d1 ^ 2 = 3072 + 6728;
2 * d1 ^ 2 = 9800;
d1 ^ 2 = 9800/2;
d1 ^ 2 = 4900;
d1 = √4900;
d1 = 70 cm.
Answer: d1 = 70 cm.