Let's determine the coordinates of point [tex]\( G \)[/tex] which divides the line segment [tex]\( \overline{DF} \)[/tex] in the ratio [tex]\( 2:3 \)[/tex].
Given:
- Coordinates of point [tex]\( D \)[/tex] are [tex]\( (-11, 13) \)[/tex].
- Coordinates of point [tex]\( F \)[/tex] are [tex]\( (4, 8) \)[/tex].
- The ratio of [tex]\( DG : GF = 2 : 3 \)[/tex].
To find the coordinates of [tex]\( G \)[/tex], we use the section formula for internal division of a line segment.
Firstly, recall the section formula:
If a point [tex]\( P(x, y) \)[/tex] divides the line segment joining points [tex]\( A(x_1, y_1) \)[/tex] and [tex]\( B(x_2, y_2) \)[/tex] in the ratio [tex]\( m:n \)[/tex], then:
[tex]\[ P(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \][/tex]
Here, our points [tex]\( A(x_1, y_1) \)[/tex] and [tex]\( B(x_2, y_2) \)[/tex] are [tex]\( D(-11, 13) \)[/tex] and [tex]\( F(4, 8) \)[/tex] respectively. The ratio [tex]\( m:n \)[/tex] is [tex]\( 2:3 \)[/tex].
Let's substitute these values into the section formula to find the coordinates of [tex]\( G \)[/tex].
[tex]\[
G_x = \frac{(2 \cdot 4) + (3 \cdot -11)}{2 + 3} = \frac{8 - 33}{5} = \frac{-25}{5} = -5
\][/tex]
[tex]\[
G_y = \frac{(2 \cdot 8) + (3 \cdot 13)}{2 + 3} = \frac{16 + 39}{5} = \frac{55}{5} = 11
\][/tex]
Thus, the coordinates of [tex]\( G \)[/tex] are [tex]\( (-5, 11) \)[/tex].
So, the coordinates of point [tex]\( G \)[/tex] are [tex]\(\boxed{(-5, 11)}\)[/tex].
