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The question in the image asks to determine the coordinates of point \( C \) that divides the line segment \( AB \) in the ratio \( 1:3 \). The coordinates of \( A \) and \( B \) are given, and intermediate steps show how to compute the slope, the run, and the rise.

### Here's a breakdown of the steps involved:

1. **Coordinates of points**:
   - Point \( A \) is located at \( (0, 0) \).
   - Point \( B \) is located at \( (-8, 3) \).

2. **Slope of the line**:
   The slope of the line is calculated using the formula:
   \[
   \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 0}{-8 - 0} = \frac{3}{-8} = -0.375.
   \]
   This value is correctly computed in the image.

3. **Ratio and division of the segment**:
   The ratio of the division is \( 1:3 \), meaning that the point \( C \) divides the segment into two parts where one part is \( \frac{1}{4} \) (since \( \frac{1}{1+3} \)) of the total length.

4. **Run and rise**:
   - The total **run** (horizontal change) from \( B \) to \( A \) is \( 8 \) units (from \( -8 \) to \( 0 \)).
   - Taking \( \frac{1}{4} \) of the run: \( \frac{1}{4} \times 8 = 2 \). Thus, the x-coordinate of \( C \) is \( -8 + 2 = -6 \).
   - The total **rise** (vertical change) is \( 3 \) units.
   - Taking \( \frac{1}{4} \) of the rise: \( \frac{1}{4} \times 3 = 0.75 \). So, the y-coordinate of \( C \) is \( 0 + 0.75 = 0.75 \).

5. **Coordinates of point \( C \)**:
   Thus, point \( C \) is located at \( (-6, 0.75) \).

### Solution Summary:
- Point \( C \), which divides \( AB \) in the ratio \( 1:3 \), is located at \( (-6, 0.75) \).

Would you like more details on any step or a deeper explanation?

Here are five related questions to explore:
1. How is the concept of slope useful in determining points along a line?
2. What are other methods to divide a line segment into a given ratio?
3. Can you apply the midpoint formula to verify the division of the segment?
4. How do you generalize this to find a point dividing any two points in a ratio \( m:n \)?
5. What would the coordinates be if the ratio was reversed, say \( 3:1 \)?

**Tip:** The formula to divide a line segment in the ratio \( m:n \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[
\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right).
\]

Determine the coordinates of point C that divides AB into two segments in the ratio 1:3.

Learn how to calculate the coordinates of point C, which divides line segment AB into a 1:3 ratio using coordinate geometry. The solution involves determining the slope, run, rise, and applying the line division formula. Suitable for students in Grades 9-11.

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