In a rectangular trapezoid, the sides are 12 cm and 20 cm, and the smaller diagonal is 13 cm. Find the base of the trapezoid.

From the right-angled triangle ABC, according to the Pythagorean theorem, we determine the length of the leg BC.

BC ^ 2 = AC ^ 2 – AB ^ 2 = 13 ^ 2 – 12 ^ 2 = 169 – 144 = 25.

BC = 5 cm.

Let’s draw the height of the CH trapezoid. Since ABCН is a rectangle, the length of CH = AB = 12 cm, AH = BH = 5 cm.

In a right-angled triangle СDН, we define, according to the Pythagorean theorem, the length of the leg DH.

DH ^ 2 = CD ^ 2 – CH ^ 2 = 20 ^ 2 – 12 ^ 2 = 400 – 144 = 256.

DН = 16 cm.

Then the length of the segment АD = АН + DH = 5 + 16 = 21 cm.

Answer: The lengths of the bases of the trapezoid are 5 cm and 21 cm.