To find the coordinates of point [tex]\( G \)[/tex] which divides the directed line segment [tex]\( \overline{FH} \)[/tex], we use the section formula. The section formula helps determine a point that divides a line segment in a given ratio, either internally or externally. In this case, the ratio given is [tex]\( 6 : 2 \)[/tex].
Given points:
[tex]\[ F(-10, -4) \][/tex]
[tex]\[ H(8, 4) \][/tex]
And the ratio [tex]\( m : n = 6 : 2 \)[/tex].
The section formula for internal division is:
[tex]\[ G = \left( \frac{m \cdot x_2 + n \cdot x_1}{m + n}, \frac{m \cdot y_2 + n \cdot y_1}{m + n} \right) \][/tex]
Substituting the given values:
[tex]\[ x_1 = -10, \quad y_1 = -4 \][/tex]
[tex]\[ x_2 = 8, \quad y_2 = 4 \][/tex]
[tex]\[ m = 6, \quad n = 2 \][/tex]
First, we find the sum of the ratio terms:
[tex]\[ m + n = 6 + 2 = 8 \][/tex]
Now we calculate the x-coordinate of [tex]\( G \)[/tex]:
[tex]\[ G_x = \frac{(6 \cdot 8) + (2 \cdot -10)}{8} = \frac{48 + (-20)}{8} = \frac{28}{8} = 3.5 \][/tex]
Next, we calculate the y-coordinate of [tex]\( G \)[/tex]:
[tex]\[ G_y = \frac{(6 \cdot 4) + (2 \cdot -4)}{8} = \frac{24 + (-8)}{8} = \frac{16}{8} = 2.0 \][/tex]
Therefore, the coordinates of point [tex]\( G \)[/tex] that divides the segment [tex]\( \overline{FH} \)[/tex] in the ratio 6:2 are:
[tex]\[ G = (3.5, 2.0) \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{(3.5, 2)} \][/tex]
