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

To find the coordinates of a point that partitions a directed line segment in a given ratio, you can use the section formula. If a point \( P(x, y) \) divides the line segment joining \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \), then the coordinates of \( P \) are given by:

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

In your case, the coordinates of points \( A \) and \( B \) are:
\[
A(-6, -4) \quad \text{and} \quad B(3, -1)
\]
The ratio \( m:n \) is \( 1:2 \).

Substituting the values into the section formula:

\[
x = \frac{1 \cdot 3 + 2 \cdot (-6)}{1+2} = \frac{3 - 12}{3} = \frac{-9}{3} = -3
\]
\[
y = \frac{1 \cdot (-1) + 2 \cdot (-4)}{1+2} = \frac{-1 - 8}{3} = \frac{-9}{3} = -3
\]

So, the coordinates of the point are \( \boxed{(-3, -3)} \).

Would you like more details or have any questions?

Here are 5 related questions to further your understanding:

1. How do you find the midpoint of a line segment?
2. How does the section formula change when dividing externally?
3. What is the significance of the directed line segment in geometry?
4. How would the coordinates change if the ratio was reversed to 2:1?
5. Can you apply the section formula to three-dimensional coordinates?

**Tip:** The section formula can also be used to find points dividing a segment in a negative ratio, which corresponds to external division.

Learn how to find the coordinates of a point using the section formula in geometry. This problem involves determining the point that partitions a directed line segment from (-6, -4) to (3, -1) in a ratio of 1:2. Detailed step-by-step solution provided.

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