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

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To solve the problem, we need to find the value of $$ x $$ and the length of each diagonal of rectangle $$ LMNP $$.

### Step 1: Set the sides equal
In a rectangle, opposite sides are equal. Therefore, we can set the expressions for $$ LN $$ and $$ MP $$ equal to each other:

$$
12x - 6 = 7x + 7
$$

### Step 2: Solve for $$ x $$
Now, we will solve the equation for $$ x $$.

1. Subtract $$ 7x $$ from both sides:
   $$
   12x - 7x - 6 = 7
   $$
   This simplifies to:
   $$
   5x - 6 = 7
   $$

2. Add $$ 6 $$ to both sides:
   $$
   5x = 13
   $$

3. Divide by $$ 5 $$:
   $$
   x = \frac{13}{5}
   $$

### Step 3: Find the length of $$ LN $$
Now that we have $$ x $$, we can substitute it back into the expression for $$ LN $$:

$$
LN = 12x - 6
$$

Substituting $$ x = \frac{13}{5} $$:

$$
LN = 12\left(\frac{13}{5}\right) - 6
$$

Calculating $$ 12 \times \frac{13}{5} $$:

$$
LN = \frac{156}{5} - 6
$$

Convert $$ 6 $$ to a fraction with a denominator of $$ 5 $$:

$$
6 = \frac{30}{5}
$$

Now, subtract:

$$
LN = \frac{156}{5} - \frac{30}{5} = \frac{126}{5}
$$

### Step 4: Find the length of each diagonal
In a rectangle, the length of the diagonal $$ d $$ can be calculated using the Pythagorean theorem:

$$
d = \sqrt{LN^2 + MP^2}
$$

Since $$ LN = MP $$, we can use $$ LN $$ to find the diagonal:

$$
d = \sqrt{LN^2 + LN^2} = \sqrt{2 \cdot LN^2} = LN \sqrt{2}
$$

Substituting $$ LN = \frac{126}{5} $$:

$$
d = \frac{126}{5} \sqrt{2}
$$

### Final Answers
- The value of $$ x $$ is $$ \frac{13}{5} $$.
- The length of $$ LN $$ is $$ \frac{126}{5} $$.
- The length of each diagonal is $$ \frac{126}{5} \sqrt{2} $$.

Thus, the final answers are:
- $$ x = \frac{13}{5} $$
- $$ LN = \frac{126}{5} $$
$$ x = \frac{13}{5} $$ and $$ LN = \frac{126}{5} $$.

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

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