## Geometry: Common Core (15th Edition)

$x = \frac{5}{2}$ $LN = \frac{25}{2}$ $MP = \frac{25}{2}$
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: $3x + 5 = 9x - 10$ Subtract $9x$ from each side of the equation to move variables to the left side of the equation: $-6x + 5 = -10$ Subtract $5$ from each side of the equation to move constants to the right side of the equation: $-6x = -15$ Divide each side by $-6$ to solve for $x$: $x = \frac{15}{6}$ Divide the numerator and denominator by their greatest common factor, $3$: $x = \frac{5}{2}$ Now that we have the value of $x$, we can plug $\frac{5}{2}$ in for $x$: $LN = 3(\frac{5}{2}) + 5$ Multiply first, according to order of operations: $LN = \frac{15}{2} + 5$ Convert $5$ into a fraction that has $2$ as its denominator: $LN = \frac{15}{2} + \frac{10}{2}$ Subtract to solve: $LN = \frac{25}{2}$ Now let's find $MP$: $MP = 9(\frac{5}{2}) - 10$ Multiply first, according to order of operations: $MP = \frac{45}{2} - 10$ Convert $10$ into a fraction with $2$ as its denominator: $MP = \frac{45}{2} - \frac{20}{2}$ Add to solve: $MP = \frac{25}{2}$