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.)

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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.
$$ x = \frac{13}{5} $$ and each diagonal is approximately 35.64 units long.

LMNP is a rectangle. Find the value of and the length of each diagonal.

(Simplify your answer.)

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LMNP is a rectangle. Find the value of and the length of each diagonal. (Simplify your answer.)