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To find the point that divides the line segment from point A to point B in a 1:4 ratio, we will use the section formula. The coordinates of A and B are as follows:

- \( A = (-4, -4) \)
- \( B = (6, 6) \)

### Section Formula:
If a point divides the line segment joining two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \), the coordinates of the point are given by:

\[
P = \left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n} \right)
\]

In this case:
- \( m = 1 \)
- \( n = 4 \)
- \( x_1 = -4 \), \( y_1 = -4 \) (coordinates of A)
- \( x_2 = 6 \), \( y_2 = 6 \) (coordinates of B)

Now, applying the formula:

\[
P_x = \frac{1 \cdot 6 + 4 \cdot (-4)}{1+4} = \frac{6 - 16}{5} = \frac{-10}{5} = -2
\]

\[
P_y = \frac{1 \cdot 6 + 4 \cdot (-4)}{1+4} = \frac{6 - 16}{5} = \frac{-10}{5} = -2
\]

Thus, the point that divides the line segment in the 1:4 ratio is \( (-2, -2) \).

Would you like more details or have any questions?

### Related Questions:
1. How would the coordinates change if the ratio was 2:3 instead of 1:4?
2. What is the distance between points A and B?
3. Can you find the midpoint of the line segment AB?
4. How would you generalize the section formula for a 3D coordinate system?
5. What would be the coordinates if the ratio was reversed (4:1)?

### Tip:
Always ensure you assign the correct values for the ratio and the coordinates of the points when applying the section formula!

What is the point that divides the directed line segment from point A to point B in a 1:4 ratio?

Learn how to find the point that divides a directed line segment from point A to point B in a 1:4 ratio using the section formula. This detailed step-by-step guide covers key concepts in coordinate geometry, ratios, and the section formula, making it suitable for students in Grades 9-12.

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