Area of trapezoid ABCD?

A hard-level "Area" problem.It is no accident that 9 and 25 are square numbers. You can jump straight to (3+5)² = 64.

Hints

AD and BC are parallel. Squeezed between two parallel lines, the upper triangle ADE and the lower triangle CBE are similar — same shape, different size.
For similar shapes, the ratio of areas is "side ratio × side ratio". 9 : 25 is 3×3 : 5×5, so the side ratio is 3 : 5, i.e. AE : EC = 3 : 5.
Look at triangle ABE and triangle CBE. They share vertex B, and their bases AE and EC lie on the same line, so they have the same height. Triangles with the same height have areas proportional to their bases, so the area ratio is also 3 : 5.
Triangle CBE is 25 cm² and the ratio is 3 : 5. 25 ÷ 5 = 5, 5 × 3 = 15. So triangle ABE = 15 cm².
Now triangle ADE and triangle DEC, which share vertex D. For exactly the same reason the area ratio is the same.
Triangle ADE is 9 cm² and the ratio is 3 : 5. 9 ÷ 3 = 3, 3 × 5 = 15. So triangle DEC = 15 cm².
All four areas are known. Add them up to get the trapezoid. 9 + 15 + 15 + 25

Answer 64cm²