$$ \overline{P Q} $$ is bisected at $$ R $$. If $$ P R=5 x+54 $$ and $$ R Q=x+14 $$, what is the length of $$ \overline{P Q} $$ ?

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To determine the length of $$ \overline{P Q} $$, let's follow these steps:

1. **Understand the Given Information:**
   - $$ R $$ is the midpoint of $$ \overline{P Q} $$, so $$ PR = RQ $$.
   - $$ PR = 5x + 54 $$
   - $$ RQ = x + 14 $$

2. **Set Up the Equation:**
   Since $$ PR = RQ $$,
   $$
   5x + 54 = x + 14
   $$

3. **Solve for $$ x $$:**
   $$
   5x + 54 = x + 14 \
   5x - x = 14 - 54 \
   4x = -40 \
   x = -10
   $$

4. **Find $$ PR $$ and $$ RQ $$ Using $$ x $$:**
   $$
   PR = 5(-10) + 54 = -50 + 54 = 4 \
   RQ = -10 + 14 = 4
   $$

5. **Determine the Length of $$ \overline{P Q} $$:**
   Since $$ R $$ is the midpoint, $$ PQ = 2 \times PR $$ or $$ PQ = PR + RQ $$:
   $$
   PQ = 2 \times 4 = 8 \quad \text{or} \quad PQ = 4 + 4 = 8
   $$

**Final Answer:**  
The length of $$ \overline{P Q} $$ is **8 units**.
The length of $$ \overline{P Q} $$ is 8 units.

\( \overline{P Q} \) is bisected at \( R \). If \( P R=5 x+54 \) and \( R Q=x+14 \), what is the length of \( \overline{P Q} \) ?

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