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The problem asks for the coordinates of point \( H \) on the line segment \( FG \), where the coordinates of \( F \) are \( (-8, 3) \) and those of \( G \) are \( (-2, -9) \). The point \( H \) divides the segment such that the ratio \( FH : HG = 1:2 \).

To solve this, we'll use the section formula, which is given as:

\[
H = \left( \frac{m \cdot x_2 + n \cdot x_1}{m + n}, \frac{m \cdot y_2 + n \cdot y_1}{m + n} \right)
\]

Where:
- \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of \( F \) and \( G \),
- \( m:n \) is the ratio \( 1:2 \).

Substituting the values:
- \( (x_1, y_1) = (-8, 3) \) (coordinates of \( F \)),
- \( (x_2, y_2) = (-2, -9) \) (coordinates of \( G \)),
- \( m = 1 \) and \( n = 2 \).

Let’s compute this.The coordinates of point \( H \) are \( (-6, -1) \).

Would you like further details or have any questions? Here are some related questions for further exploration:

1. How does the section formula change when the ratio is reversed, like \( 2:1 \)?
2. How would the coordinates change if the ratio were \( 3:1 \)?
3. Can the section formula be applied to three-dimensional points?
4. What if \( H \) were the midpoint of the segment? What would the coordinates be?
5. How would you determine the distance between points \( F \), \( H \), and \( G \)?

**Tip**: The section formula can be a powerful tool for dividing line segments in any ratio, both internally and externally!

The coordinates of the endpoints of FG are F(-8, 3) and G(-2, -9). Point H is on FG and divides it such that FH:GH is 1:2. What are the coordinates of H?

Solve for the coordinates of point H that divides the line segment FG in a 1:2 ratio using the section formula. The endpoints are F(-8, 3) and G(-2, -9). Suitable for Grades 9-10 students learning coordinate geometry.

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