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We are given two points, \( A(-5, 3) \) and \( B(7, -9) \), and we are asked to find the coordinates of the point that divides the line segment \( AB \) in the ratio \( 2:3 \).

### Solution:

To find the point that divides the line segment \( AB \) in a given ratio, we use the **section formula**. The formula for a point that divides a segment \( AB \) with coordinates \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \) is:

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

Given:
- \( A(-5, 3) \)
- \( B(7, -9) \)
- Ratio \( m:n = 2:3 \)

#### Applying the formula:

\[
x = \frac{2 \cdot 7 + 3 \cdot (-5)}{2 + 3} = \frac{14 - 15}{5} = \frac{-1}{5} = -0.2
\]

\[
y = \frac{2 \cdot (-9) + 3 \cdot 3}{2 + 3} = \frac{-18 + 9}{5} = \frac{-9}{5} = -1.8
\]

So, the coordinates of the point that divides the line segment \( AB \) in the ratio \( 2:3 \) are:

\[
\boxed{(-0.2, -1.8)}
\]

Let me know if you'd like more details on this solution or have any questions!

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Here are five related questions to further explore the concept:
1. How would the formula change if the ratio was \( 3:2 \) instead of \( 2:3 \)?
2. What if the point divided the segment externally in the ratio \( 2:3 \)?
3. How can we find the midpoint of the segment \( AB \)?
4. How does the section formula apply in three-dimensional geometry?
5. What if the division ratio were negative?

**Tip:** Always ensure the ratio is positive when dealing with internal division and negative for external division.

Diberikan titik-titik A(-5, 3) dan B(7, -9). Tentukan koordinat titik yang membagi segmen AB dengan perbandingan 2:3.

Learn how to find the point that divides a line segment with endpoints A(-5, 3) and B(7, -9) in the ratio 2:3. This detailed solution uses the section formula for internal division. Ideal for students in Grades 9-10.

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