To solve this problem, we need to determine the coordinates of point $Q$ on the segment $PR$, such that $PQ$ is $\frac{5}{7}$ of $PR$.

Step-by-step solution:

1. Given Data:

   • Point $P(-29, 12)$
   • Point $R(6, -2)$
   • $Q$ divides $PR$ in the ratio $5:2$ since $\frac{PQ}{PR} = \frac{5}{7}$.

2. Formula for section division: If a point $Q(x, y)$ divides a line segment $PR$ in the ratio $m:n$, the coordinates of $Q$ are: $Q\left( \frac{m \cdot x_2 + n \cdot x_1}{m+n}, \frac{m \cdot y_2 + n \cdot y_1}{m+n} \right)$ Here:

   • $P(x_1, y_1) = (-29, 12)$
   • $R(x_2, y_2) = (6, -2)$
   • $m = 5$, $n = 2$.

3. Plug in values:

   • For $x$-coordinate of $Q$: $x_Q = \frac{5 \cdot 6 + 2 \cdot (-29)}{5+2} = \frac{30 - 58}{7} = \frac{-28}{7} = -4$

   • For $y$-coordinate of $Q$: $y_Q = \frac{5 \cdot (-2) + 2 \cdot 12}{5+2} = \frac{-10 + 24}{7} = \frac{14}{7} = 2$

4. Final Coordinates: $Q(-4, 2)$

Would you like additional details or help with another question? Here are related questions to deepen your understanding:

1. How do you derive the section formula for dividing a line segment in a ratio?
2. What is the midpoint of the segment $PR$?
3. How would the solution change if $Q$ divided $PR$ externally in the same ratio?
4. What is the length of $PQ$ and $QR$?
5. Can you verify $\frac{PQ}{PR} = \frac{5}{7}$ using the coordinates?

Tip: When using the section formula, always confirm whether the division is internal or external for accuracy.