Calculate DC Length in Parallelogram: Using AE/DC = 1/2 Ratio

To solve this problem, we'll follow these steps:

• Step 1: Understand that $\frac{AE}{DC} = \frac{1}{2}$ implies $AE = \frac{1}{2} \times DC$.

• Step 2: Use $DC$ as the base of the parallelogram and express the height in terms of $DC$.

• Step 3: Use the area formula: $\text{Area} = \text{base} \times \text{height}$.

• Step 4: Solve for $DC$, knowing the total area is 98 cm².

Now, let's work through each step:
Step 1: Given $\frac{AE}{DC} = \frac{1}{2}$, we can express $AE$ as $\frac{1}{2} \times DC$.

Step 2: Assume $DC$ is the base, and $AE$ as a related height gives $\text{height} = \frac{1}{2} \times DC$.

Step 3: Since $\text{Area} = \text{base} \times \text{height}$, substitute $DC$ for the base and $\frac{1}{2} \times DC$ for the height:
$98 = DC \times \left( \frac{1}{2} \times DC \right)$

Step 4: Simplify and solve for $DC$: $98 = \frac{1}{2} \times DC^2$. This simplifies to:
multiply both sides by 2
$196 = DC^2$
take the square root
$DC = \sqrt{196} = 14 \, \text{cm}$

Therefore, the length of $DC$ is $\text{14 cm}$.