Answer
$x = 9$
$LN = 67$
$MP = 67$
Work Step by Step
According to Theorem 6-15, the diagonals of a rectangle are congruent. Therefore, we can set $LN$ and $MP$, the diagonals of $LMNP$, equal to one another to solve for $x$:
$LN = MP$
Substitute with the expressions given for each diagonal:
$9x - 14 = 7x + 4$
Subtract $7x$ from each side of the equation to move variables to the left side of the equation:
$2x - 14 = 4$
Add $14$ to each side of the equation to move constants to the right side of the equation:
$2x = 18$
Divide each side by $2$ to solve for $x$:
$x = 9$
Now that we have the value of $x$, we can plug $9$ in for $x$ into the expressions for each diagonal to find the length of each diagonal.
$LN = 9(9) - 14$
Multiply first, according to order of operations:
$LN = 81 - 14$
Subtract to solve:
$LN = 67$
Now let's find $MP$:
$MP = 7(9) + 4$
Multiply first, according to order of operations:
$MP = 63 + 4$
Add to solve:
$MP = 67$
