Answer
$x = \frac{5}{3}$
$LN = \frac{29}{3}$
$MP = \frac{29}{3}$
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:
$7x - 2 = 4x + 3$
Subtract $4x$ from each side of the equation to move variables to the left side of the equation:
$3x - 2 = 3$
Add $2$ to each side of the equation to move constants to the right side of the equation:
$3x = 5$
Divide each side by $3$ to solve for $x$:
$x = \frac{5}{3}$
Now that we have the value of $x$, we can plug $\frac{5}{3}$ in for $x$:
$LN = 7(\frac{5}{3}) - 2$
$LN = \frac{35}{3} - 2$
$LN = \frac{35}{3} - \frac{6}{3}$
$LN = \frac{29}{3}$
Now let's find $MP$:
$MP = 4(\frac{5}{3}) + 3$
Multiply first, according to order of operations:
$MP = \frac{20}{3} + 3$
Convert $3$ into a fraction with $3$ as its denominator:
$MP = \frac{20}{3} + \frac{9}{3}$
Add to solve:
$MP = \frac{29}{3}$
