Trapezoid Base Lengths: Solving for Bases Given Area=30cm² and Height=5cm

To solve this problem, we'll use the following plan:

• Identify the given values: The area is $30 \, \text{cm}^2$, and the height is $5 \, \text{cm}$.
• Use the area formula for a trapezoid, which is $\text{Area} = \frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{Height}$.
• Recognize the relationship: Since AB is half the length of DC, let the length of AB be $x$ and DC be $2x$.
• Substitute the known values and relationships into the formula to find $x$.

Let's proceed with the calculation:

The formula for the area of a trapezoid is:

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

Substitute the known values:

$30 = \frac{1}{2} \times (x + 2x) \times 5$

Combine the bases:

$30 = \frac{1}{2} \times 3x \times 5$

Simplify: $30 = \frac{15x}{2}$

Multiply both sides by 2 to clear the fraction:

$60 = 15x$

Divide both sides by 15:

$x = 4$

Therefore, the bases are:

• AB (the shorter base) is $4 \, \text{cm}$.
• DC (the longer base) is $2x = 2 \times 4 = 8 \, \text{cm}$.

The lengths of the bases of the trapezoid are $\textbf{4 cm and 8 cm.}$