Answer
a. $x = 7$
b. Diagonals $AC$ and $DB$ are $9$.
Work Step by Step
a. The diagram is that of an isosceles trapezoid; therefore, the diagonals are congruent to one another.
The diagonals of this trapezoid are $\overline{DB}$ and $\overline{AC}$. Set the two diagonals equal to one another:
$\overline{DB}$ = $\overline{AC}$
Plug in what we know:
$(2x - 8) + (x - 4) = x + 2$
Rewrite the equation without parentheses:
$2x - 8 + x - 4 = x + 2$
Combine like terms on the left side of the equation:
$3x - 12 = x + 2$
Subtract $x$ from both sides of the equation to move variables to the left side of the equation:
$2x - 12 = 2$
Add $12$ to both sides of the equation to move constants to the right side of the equation:
$2x = 14$
Divide each side by $2$ to solve for $x$:
$x = 7$
b. Let's plug in $x$ into the expression for $\overline{AC}$ to find the length of this diagonal:
$AC = 7 + 2$
Add to solve:
$AC = 9$
Since the two diagonals are congruent to each other, $DB$ is also $9$.
