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To find point \( G \) on line segment \( AB \) such that the ratio of \( AG \) to \( GB \) is 3:2, we need to apply the section formula, which helps find a point dividing a line segment in a given ratio.

### Steps to follow:

1. **Find the coordinates of points A and B** from the graph:
   - Point \( A \): \( (-3, 4) \)
   - Point \( B \): \( (5, 0) \)

2. **Use the section formula**:
   The section formula in a 2D plane for a point \( G(x, y) \) dividing a line segment joining points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \) is:
   \[
   G(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
   \]
   Here, \( m:n = 3:2 \), and the coordinates of \( A \) and \( B \) are \( (-3, 4) \) and \( (5, 0) \), respectively.

3. **Apply the formula**:
   Substituting \( x_1 = -3, y_1 = 4, x_2 = 5, y_2 = 0, m = 3, n = 2 \), we get:
   \[
   G(x, y) = \left( \frac{3(5) + 2(-3)}{3+2}, \frac{3(0) + 2(4)}{3+2} \right)
   \]
   Simplifying:
   \[
   G(x, y) = \left( \frac{15 - 6}{5}, \frac{0 + 8}{5} \right)
   \]
   \[
   G(x, y) = \left( \frac{9}{5}, \frac{8}{5} \right)
   \]
   So, the coordinates of point \( G \) are \( G\left( \frac{9}{5}, \frac{8}{5} \right) \).

Thus, point \( G \) divides line segment \( AB \) in the ratio 3:2 at coordinates \( \left( \frac{9}{5}, \frac{8}{5} \right) \), or approximately \( (1.8, 1.6) \).

Would you like more details on this process, or do you have further questions?

Here are 5 related questions you can explore:
1. How do we find the midpoint of a line segment using coordinates?
2. Can the section formula be used in three dimensions?
3. What is the importance of the ratio in determining the point's position?
4. How can we verify that point G divides \( AB \) in the correct ratio?
5. What other methods can be used to find a dividing point without using the section formula?

**Tip**: Always ensure the ratio is properly applied to both x- and y-coordinates separately when using the section formula.

Find point G on AB such that the ratio of AG to GB is 3:2.

This problem involves using the section formula to find a point G on line AB that divides the line in the ratio 3:2. The solution involves applying the section formula for coordinates, identifying points A and B from a given graph, and calculating the exact position of G.

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