To find the coordinates of point G that lies on the line segment FH, such that the ratio of FG to FH is 1:6, we can use the concept of section formula. The section formula helps to find the coordinates of a point that divides a line segment into a given ratio.
### Step-by-Step Solution
1. Identify the coordinates of the given points:
- F has coordinates [tex]\( F(4, -5) \)[/tex]
- H has coordinates [tex]\( H(-8, 19) \)[/tex]
2. Understand the ratio:
- The ratio of FG to FH is given as [tex]\( 1:6 \)[/tex]. This means point G divides the line segment FH in the ratio 1:5 internally (since FG is 1 part and GH is 5 parts out of the total 6 parts).
3. Use the section formula:
- The section formula for the coordinates [tex]\((x, y)\)[/tex] of the point G dividing the segment FH in the ratio [tex]\(m:n\)[/tex] is:
[tex]\[
G_x = \frac{mx_2 + nx_1}{m + n}
\][/tex]
[tex]\[
G_y = \frac{my_2 + ny_1}{m + n}
\][/tex]
where [tex]\( (x_1, y_1) \)[/tex] are the coordinates of point F and [tex]\( (x_2, y_2) \)[/tex] are the coordinates of point H, and [tex]\( m = 1 \)[/tex] and [tex]\( n = 5 \)[/tex] in this case.
4. Substitute the values into the formula:
- Substitute [tex]\(m = 1\)[/tex], [tex]\(n = 5\)[/tex], [tex]\(x_1 = 4\)[/tex], [tex]\(y_1 = -5\)[/tex], [tex]\(x_2 = -8\)[/tex], and [tex]\(y_2 = 19\)[/tex].
5. Calculate the x-coordinate of G:
[tex]\[
G_x = \frac{1 \cdot (-8) + 5 \cdot 4}{1 + 5} = \frac{-8 + 20}{6} = \frac{12}{6} = 2
\][/tex]
6. Calculate the y-coordinate of G:
[tex]\[
G_y = \frac{1 \cdot 19 + 5 \cdot (-5)}{1 + 5} = \frac{19 - 25}{6} = \frac{-6}{6} = -1
\][/tex]
### Conclusion
Hence, the coordinates of point G that divides the line segment FH in the ratio 1:6 are [tex]\((2.0, -1.0)\)[/tex].
