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The question asks to find the coordinates of a point on the directed line segment from \((-6, -2)\) to \((9, -7)\) that divides the segment in the ratio 3:2.

To solve this, we can use the section formula (or ratio formula). The section formula for a point dividing the line segment joining \((x_1, y_1)\) and \((x_2, y_2)\) in the ratio \(m:n\) is:

\[
\left(\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}\right)
\]

In this case, \((x_1, y_1) = (-6, -2)\), \((x_2, y_2) = (9, -7)\), and the ratio \(m:n = 3:2\).

Let’s calculate the coordinates.

### For the \(x\)-coordinate:
\[
x = \frac{(3 \cdot 9) + (2 \cdot (-6))}{3 + 2} = \frac{27 - 12}{5} = \frac{15}{5} = 3
\]

### For the \(y\)-coordinate:
\[
y = \frac{(3 \cdot (-7)) + (2 \cdot (-2))}{3 + 2} = \frac{-21 - 4}{5} = \frac{-25}{5} = -5
\]

Thus, the coordinates of the point that divides the line segment in the ratio 3:2 are \((3, -5)\).

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Would you like more details on this solution or have any questions?

Here are 5 related questions:
1. How would the solution change if the ratio was 2:3 instead of 3:2?
2. What is the formula for the midpoint of a line segment?
3. How can you calculate the length of a line segment given two endpoints?
4. Can the section formula be used for 3D coordinates as well? How would it differ?
5. What happens if the ratio is negative? How does that affect the section formula?

**Tip**: The section formula can also be used to divide a line externally, not just internally.

What are the coordinates of the point on the directed line segment from (-6, -2) to (9, -7) that partitions the segment into a ratio of 3:2?

Learn how to find the coordinates of the point that partitions the line segment from (-6, -2) to (9, -7) into a 3:2 ratio using the section formula. This detailed step-by-step solution covers important concepts in coordinate geometry and ratio division, suitable for high school students.

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