To solve the problem, let's analyze it step-by-step:

1. Given Information:
   $\triangle MNP \sim \triangle QRP$.
   This means the two triangles are similar, so corresponding sides are proportional.

2. Corresponding Sides:
   From the similarity, we can set up proportions: $\frac{MN}{QR} = \frac{NP}{RP} = \frac{MP}{QP}.$

   Substituting the given lengths:

   • $MN = x + 8$, $QR = 28$,
   • $NP = 24$, $RP = 3x - 9$,
   • $MP = 24$, $QP = 28$.

3. Set up the Proportion:
   From $\frac{NP}{RP} = \frac{MP}{QP}$, substitute the values: $\frac{24}{3x - 9} = \frac{24}{28}.$

4. Simplify the Proportion:
   The fraction on the right simplifies to: $\frac{24}{3x - 9} = \frac{6}{7}.$

5. Solve for $x$:
   Cross-multiply: $24 \cdot 7 = 6 \cdot (3x - 9),$ $168 = 18x - 54.$ Simplify: $18x = 222,$ $x = 12.33 \, (\text{or } \frac{37}{3}).$

6. Find $PR$:
   Substitute $x = 12.33$ into $RP = 3x - 9$: $RP = 3(12.33) - 9 = 36.99 - 9 = 27.99 \, (\text{approximately } 28).$

Thus, $PR \approx 28$.

Would you like a more detailed explanation of any step or further assistance? Here are some related questions to explore:

1. How are similar triangles identified?
2. Why is the property of proportionality crucial for similar triangles?
3. What happens if one triangle’s side lengths are scaled incorrectly?
4. Can this problem be solved using an alternative approach, such as solving for all unknown sides first?
5. How can you verify the solution is consistent with all proportions?

Tip: Always simplify fractions before solving equations—it often reduces calculation errors!