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The problem asks for the coordinates of a point that partitions a directed line segment in a ratio of 1 to 2. The line segment extends from the point \( A(-6, -4) \) to the point \( B(3, -1) \), and the ratio is given as \( 1:2 \).

To solve this, we can use the section formula, which gives the coordinates of a point dividing a line segment in a given ratio. The formula is:

\[
P(x, y) = \left( \frac{m_1x_2 + m_2x_1}{m_1 + m_2}, \frac{m_1y_2 + m_2y_1}{m_1 + m_2} \right)
\]

Where:
- \( P(x, y) \) are the coordinates of the point dividing the line,
- \( (x_1, y_1) \) are the coordinates of point \( A \),
- \( (x_2, y_2) \) are the coordinates of point \( B \),
- \( m_1 : m_2 \) is the given ratio.

For this problem:
- \( A(-6, -4) \),
- \( B(3, -1) \),
- The ratio \( 1:2 \), so \( m_1 = 1 \) and \( m_2 = 2 \).

Now, let’s calculate the coordinates of the point \( P \).The coordinates of the point that partitions the line segment from \( (-6, -4) \) to \( (3, -1) \) in the ratio of 1:2 are \( (-3.0, -3.0) \).

Would you like more details or have any questions?

Here are some related questions you might find helpful:
1. How is the section formula derived?
2. What is the midpoint formula, and how is it related to the section formula?
3. How would the result change if the ratio were reversed to 2:1?
4. Can this formula be applied to three-dimensional points?
5. How do we handle situations where the ratio involves negative numbers?

**Tip:** The section formula can be generalized for any number of dimensions, making it useful in a variety of geometric contexts.

What are the coordinates of the point on the directed line segment from (-6, -4) to (3, -1) that partitions the segment into a ratio of 1 to 2?

Calculate the coordinates of a point that divides the directed line segment from (-6, -4) to (3, -1) in a ratio of 1 to 2 using the section formula. This problem demonstrates key concepts of coordinate geometry, ratios, and the section formula.

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