Characteristics and Proofs of a Parallelogram: Calculating the length of a side in a parallelogram

To solve this problem, let's begin by identifying the relationship between the sides:

Since $AB = 7x - 4$ and $DC = 5x + 12$, and we know that opposite sides of a parallelogram are equal, we have:

$7x - 4 = 5x + 12$

First, let's solve for $x$:

$7x - 5x = 12 + 4$
$2x = 16$

Divide both sides by 2:

$x = 8$

Now that we have the value of $x$, substitute it back into the expression for $DC$ to find its length:

$DC = 5x + 12 = 5(8) + 12$
$DC = 40 + 12 = 52$

Therefore, the length of side $DC$ is $\mathbf{52}$.