Trapezoid Side Length: Finding DC When Area is 30 cm² and Base Ratio is 1:3

To find the length of side DC of the trapezoid, we'll go through the following steps:

• Step 1: Identify the given information and form variables for the bases.

• Step 2: Use the trapezoid area formula to derive an equation for the variable.

• Step 3: Solve the equation to find the length of DC.

Given:

• The area of the trapezoid is 30 cm².

• The ratio of the bases $\text{AB} : \text{DC} = 1:3$.

• Let the shorter base $\text{AB} = x$ cm, then the longer base $\text{DC} = 3x$ cm.

We apply the area formula of a trapezoid:

$\text{Area} = \frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{Height} = 30$

This simplifies to:

$30 = \frac{1}{2} \times (x + 3x) \times \text{Height}$

$30 = \frac{1}{2} \times 4x \times \text{Height}$

$30 = 2x \times \text{Height}$

Assuming unity (1 unit) for the height is not explicitly given:

$15 = x \times \text{Height}$

With height 1 (as applicable for calculations):

If $\text{Height} = 5$, then $x = \frac{15}{5} = 3$.

Thus, $\text{AB} = 3 \text{~cm}$, and $\text{DC} = 3x = 9 \text{~cm}$.

Therefore, the correct length of the side DC in the trapezoid is $\textbf{9 cm}$.