Calculate Trapezoid Height: Finding AE Given Area 45 cm² and Bases 4 cm, 6 cm

The problem can be solved using the formula for the area of a trapezoid:

$S = \frac{1}{2} \times (b_1 + b_2) \times h$

Where $b_1$ and $b_2$ are the lengths of the two parallel sides (bases) of the trapezoid, and $h$ is the height. For this trapezoid, the data given is:

• Area, $S = 45 \, \text{cm}^2$
• Base 1, $b_1 = 4 \, \text{cm}$
• Base 2, $b_2 = 6 \, \text{cm}$

Substitute these values into the area formula:

$45 = \frac{1}{2} \times (4 + 6) \times h$

Simplify the formula:

$45 = 5 \times h$

Divide both sides of the equation by 5 to solve for $h$:

$h = \frac{45}{5} = 9$

Thus, the length of AE, or the height of the trapezoid, is $\mathbf{9 \, \text{cm}}$.

Therefore, the solution to the problem is that the length of AE is $9 \, \text{cm}$.