LMNP is a rectangle. Find the value of \( x \) and the length of each diagonal. \( L N=12 x-6 \) and \( M P=7 x+7 \) \( x=\frac{13}{5} \) (Simplify your answer.) \( L N=\frac{126}{5} \) (Simplify your answer.) MP \( =\square \) (Simplify your answer.)

To solve for \( x \) and determine the lengths of the diagonals \( LN \) and \( MP \) in rectangle \( LMNP \), follow these steps: ### Step 1: Set Up the Equation In a rectangle, the diagonals are equal in length. Therefore, we can set the expressions for \( LN \) and \( MP \) equal to each other: \[ LN = MP \] \[ 12x - 6 = 7x + 7 \] ### Step 2: Solve for \( x \) Subtract \( 7x \) from both sides: \[ 12x - 7x - 6 = 7 \] \[ 5x - 6 = 7 \] Add \( 6 \) to both sides: \[ 5x = 13 \] Divide by \( 5 \): \[ x = \frac{13}{5} \] ### Step 3: Calculate \( LN \) Substitute \( x = \frac{13}{5} \) into the expression for \( LN \): \[ LN = 12x - 6 \] \[ LN = 12 \left( \frac{13}{5} \right) - 6 \] \[ LN = \frac{156}{5} - \frac{30}{5} = \frac{126}{5} \] ### Step 4: Calculate \( MP \) Substitute \( x = \frac{13}{5} \) into the expression for \( MP \): \[ MP = 7x + 7 \] \[ MP = 7 \left( \frac{13}{5} \right) + 7 \] \[ MP = \frac{91}{5} + \frac{35}{5} = \frac{126}{5} \] ### Summary of Results - **Value of \( x \):** \[ x = \frac{13}{5} \] - **Length of Diagonal \( LN \):** \[ LN = \frac{126}{5} \] - **Length of Diagonal \( MP \):** \[ MP = \frac{126}{5} \] Both diagonals \( LN \) and \( MP \) are equal in length, as expected in a rectangle.