LMNP is a rectangle. Find the value of \( x \) and the length of each diagonal. \[ \mathrm{LN}=12 x-6 \text { and } M P=7 x+7 \] \( x=\square \) (Simplify your answer.)

To find the value of \( x \) and the length of each diagonal in the rectangle LMNP, we can use the properties of a rectangle. Given: - \( \mathrm{LN} = 12x - 6 \) - \( \mathrm{MP} = 7x + 7 \) Since LMNP is a rectangle, opposite sides are equal in length. Therefore, we can set up the equation: \[ \mathrm{LN} = \mathrm{MP} \] Substitute the given expressions for LN and MP: \[ 12x - 6 = 7x + 7 \] Now, we can solve for \( x \) by simplifying the equation: \[ 12x - 6 = 7x + 7 \] \[ 12x - 7x = 7 + 6 \] \[ 5x = 13 \] \[ x = \frac{13}{5} \] Therefore, the value of \( x \) is \( \frac{13}{5} \). To find the length of each diagonal, we can use the Pythagorean theorem: \[ \text{Diagonal} = \sqrt{(\mathrm{LN})^2 + (\mathrm{MP})^2} \] Substitute the values of LN and MP: \[ \text{Diagonal} = \sqrt{(12x - 6)^2 + (7x + 7)^2} \] Now, we can substitute the value of \( x \) to find the length of each diagonal. Calculate the value by following steps: - step0: Calculate: \(\sqrt{\left(\frac{12\times 13}{5}-6\right)^{2}+\left(\frac{7\times 13}{5}+7\right)^{2}}\) - step1: Multiply the numbers: \(\sqrt{\left(\frac{156}{5}-6\right)^{2}+\left(\frac{7\times 13}{5}+7\right)^{2}}\) - step2: Subtract the numbers: \(\sqrt{\left(\frac{126}{5}\right)^{2}+\left(\frac{7\times 13}{5}+7\right)^{2}}\) - step3: Multiply the numbers: \(\sqrt{\left(\frac{126}{5}\right)^{2}+\left(\frac{91}{5}+7\right)^{2}}\) - step4: Add the numbers: \(\sqrt{\left(\frac{126}{5}\right)^{2}+\left(\frac{126}{5}\right)^{2}}\) - step5: Add the numbers: \(\sqrt{\frac{31752}{25}}\) - step6: Use the properties of radicals: \(\frac{\sqrt{31752}}{\sqrt{25}}\) - step7: Simplify the expression: \(\frac{126\sqrt{2}}{5}\) The length of each diagonal in the rectangle LMNP is approximately 35.64 units. Therefore, the value of \( x \) is \( \frac{13}{5} \) and the length of each diagonal is approximately 35.64 units.