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!

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.