Find point \( G \) on \( \overline{A B} \) such that the ratio of \( A G \) to \( G B \) is 3:2

To find point \( G \) on line segment \( \overline{AB} \) such that the ratio of \( AG \) to \( GB \) is \( 3:2 \), we can use the concept of dividing a segment in a given ratio. Let the length of segment \( \overline{AB} \) be denoted as \( d \). We can express the lengths \( AG \) and \( GB \) in terms of \( d \): \[ AG = \frac{3}{3+2} \cdot d = \frac{3}{5}d \] \[ GB = \frac{2}{3+2} \cdot d = \frac{2}{5}d \] Now, we can find the coordinates of point \( G \) if we know the coordinates of points \( A \) and \( B \). Assume point \( A \) has coordinates \( (x_A, y_A) \) and point \( B \) has coordinates \( (x_B, y_B) \). The coordinates of point \( G \) can be found using the section formula, which states that if a point divides a line segment in the ratio \( m:n \), then the coordinates of the point are given by: \[ G = \left( \frac{mx_B + nx_A}{m+n}, \frac{my_B + ny_A}{m+n} \right) \] In our case, \( m = 2 \) and \( n = 3 \) (since we want \( AG:GB = 3:2 \), we take \( GB \) as \( m \) and \( AG \) as \( n \)). Thus, we can substitute into the formula: \[ G = \left( \frac{2x_B + 3x_A}{2+3}, \frac{2y_B + 3y_A}{2+3} \right) \] This simplifies to: \[ G = \left( \frac{2x_B + 3x_A}{5}, \frac{2y_B + 3y_A}{5} \right) \] Thus, point \( G \) divides the segment \( \overline{AB} \) in the ratio \( 3:2 \) and its coordinates are given by the above expressions.