In an isosceles trapezoid with a lateral side of 15 cm. The smaller base is 16 cm. and the height is 9cm. find a greater foundation.

From the right-angled triangle CDН, according to the Pythagorean theorem, we determine the length of the leg of the DN.
DН ^ 2 = СD ^ 2 – CH ^ 2 = 15 ^ 2 – 9 ^ 2 = 225 – 81 = 144.
DН = 12 cm.
Let’s draw the height of the ВK trapezoid. The AВK triangle is equal to the СDН triangle in the hypotenuse and acute angle, since AB = СD, as the lateral sides of an isosceles trapezoid, the ВAK = СDН angle as angles at the base, then AK = DН = 12 cm.
The quadrilateral  ВСНK is a rectangle, since BC and СН are perpendicular to the bases of the trapezoid.
Then KН = BC = 16 cm.
AD = AK + KН + DН = 12 + 16 + 12 = 40 cm.
Answer: The larger base of the trapezoid is 40 cm.