Answer
$x = 8$
Work Step by Step
In the rectangle described, both $\overline{AC}$ and $\overline{BD}$ are diagonals of the rectangle. According to Theorem 6-15, diagonals of a rectangle are congruent to one another. Therefore, we can set the two expressions that are given for the two diagonals equal to one another:
$AC = BD$
Let's plug in our expressions for each diagonal:
$5(x - 2) = 3(x + 2)$
Use the distributive property first:
$5x - 10 = 3x + 6$
Add $10$ to each side of the equation to move constants to the right of the equation:
$5x = 3x + 16$
Subtract $3x$ from each side:
$2x = 16$
Divide each side by $2$ to solve for $x$:
$x = 8$
