Alright, let's solve this problem step by step.
Problem Statement:
On a number line, point F is at 4, and point G is at -2. Point H lies between point F and point G. The ratio of [tex]\( FH \)[/tex] to [tex]\( HG \)[/tex] is 3:9. We need to determine where point H lies on the number line.
Given Information:
- Coordinates of F: [tex]\( F = 4 \)[/tex]
- Coordinates of G: [tex]\( G = -2 \)[/tex]
- Ratio [tex]\( FH:HG = 3:9 \)[/tex]
Steps to Solve:
1. Length of the Line Segment FG:
The distance between points F and G is calculated by finding the absolute difference between the coordinates of F and G:
[tex]\[
FG = F - G = 4 - (-2) = 4 + 2 = 6
\][/tex]
2. Expressing the Ratio:
The ratio [tex]\( FH:HG = 3:9 \)[/tex]. This can be simplified to:
[tex]\[
\frac{FH}{HG} = \frac{3}{9} = \frac{1}{3}
\][/tex]
Let's denote the ratio by [tex]\( k \)[/tex]. Therefore,
[tex]\[
FH = k \cdot 3 \quad \text{and} \quad HG = k \cdot 9
\][/tex]
3. Finding the Whole Length FG:
Since [tex]\( FG = FH + HG \)[/tex]:
[tex]\[
FG = k \cdot 3 + k \cdot 9 = k \cdot (3 + 9) = k \cdot 12
\][/tex]
Given that [tex]\( FG = 6 \)[/tex], we can solve for [tex]\( k \)[/tex]:
[tex]\[
6 = k \cdot 12
\][/tex]
[tex]\[
k = \frac{6}{12} = 0.5
\][/tex]
4. Calculating FH:
Now that we have [tex]\( k = 0.5 \)[/tex], we can determine [tex]\( FH \)[/tex]:
[tex]\[
FH = 3 \cdot k = 3 \cdot 0.5 = 1.5
\][/tex]
5. Determining the Position of H:
Point H lies [tex]\( FH \)[/tex] units away from F towards G. Since [tex]\( FH = 1.5 \)[/tex]:
[tex]\[
H = F - FH = 4 - 1.5 = 2.5
\][/tex]
Therefore, point H lies at [tex]\( 2.5 \)[/tex] on the number line.
Final Answer:
Point H is at [tex]\( 2.5 \)[/tex] on the number line.
