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.)

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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.
$$ x = \frac{13}{5} $$, $$ LN = \frac{126}{5} $$, and $$ MP = \frac{126}{5} $$.

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