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We are given the directed line segment \( \overrightarrow{RT} \), and we need to find the coordinates of point \( S \) such that the ratio of \( RS \) to \( ST \) is 5:1.

### Steps to Solve:

1. **Identify the coordinates of \( R \) and \( T \):**
   - From the graph:
     - Point \( R \) is at \( (3, 2) \).
     - Point \( T \) is at \( (-7, -3) \).

2. **Use the section formula:**
   The section formula helps find a point dividing a line segment in a given ratio. For a ratio of \( m:n \), the coordinates of the point dividing the line segment joining \( (x_1, y_1) \) and \( (x_2, y_2) \) are given by:

   \[
   x = \frac{m \cdot x_2 + n \cdot x_1}{m + n}, \quad y = \frac{m \cdot y_2 + n \cdot y_1}{m + n}
   \]

   Here, the ratio \( RS:ST = 5:1 \), so \( m = 5 \) and \( n = 1 \). 

3. **Apply the formula:**
   - For the \( x \)-coordinate of \( S \):
     \[
     x_S = \frac{5 \cdot (-7) + 1 \cdot 3}{5 + 1} = \frac{-35 + 3}{6} = \frac{-32}{6} = -\frac{16}{3} \approx -5.33
     \]

   - For the \( y \)-coordinate of \( S \):
     \[
     y_S = \frac{5 \cdot (-3) + 1 \cdot 2}{5 + 1} = \frac{-15 + 2}{6} = \frac{-13}{6} \approx -2.17
     \]

So, the approximate coordinates of \( S \) are \( (-5.33, -2.17) \).

Now you can plot point \( S \) on the graph, which should be slightly left of \( x = -5 \) and slightly below \( y = -2 \).

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Would you like more details on how the section formula works or how I computed the values?

### Here are 5 questions to expand on this concept:
1. How would the section formula change if the ratio was 2:3 instead of 5:1?
2. What is the significance of a directed line segment in problems like this?
3. Can you apply the same formula to 3-dimensional coordinates?
4. What happens if the ratio \( RS:ST \) is negative?
5. How can you generalize this formula for any ratio?

**Tip:** The section formula is especially useful in coordinate geometry problems involving ratios and midpoints.

Given directed line segment RT below, find the coordinates of S such that the ratio of RS to ST is 5:1. Plot point S.

Learn how to determine the coordinates of point S dividing the line segment RT in the ratio of 5:1 using the section formula. This coordinate geometry problem helps understand the application of ratios in finding specific points.

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