Calculate Height AE in Trapezoid with Area 9x Square Centimeters

To solve this problem, we will use the formula for the area of a trapezoid:

$\text{Area} = \frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{Height}$

Here, $\text{Area} = 9x$ cm², $\text{Base}_1 = 2x$ cm, and $\text{Base}_2 = 2.5x$ cm.

Substitute the known values into the formula:

$9x = \frac{1}{2} \times (2x + 2.5x) \times \text{AE}$

$9x = \frac{1}{2} \times 4.5x \times \text{AE}$

Multiply through by 2 to clear the fraction:

$18x = 4.5x \times \text{AE}$

Solve for AE by dividing both sides by $4.5x$:

$\text{AE} = \frac{18x}{4.5x} = 4$

Thus, the height AE is $\textbf{4 cm}$.

Therefore, the solution to the problem is $\textbf{4}$ cm.