Let's solve the problem step-by-step.
1. Identify the positions of points D and E on the number line:
- Point D is at -6.
- Point E is at 8.
2. Understand the given ratio DF : FE which is 3 : 1:
This means that the total distance between D and E is divided into 3 parts for DF and 1 part for FE.
3. Calculate the total number of parts:
Since the ratio is 3 : 1, the total number of parts is [tex]\(3 + 1 = 4\)[/tex].
4. Determine the total distance between D (-6) and E (8):
To find this distance, subtract the position of D from the position of E:
[tex]\[
\text{Distance DE} = 8 - (-6) = 8 + 6 = 14
\][/tex]
5. Calculate the length of one part:
The total distance (14) divided by the total number of parts (4) gives the length of one part:
[tex]\[
\text{Length of one part} = \frac{14}{4} = 3.5
\][/tex]
6. Find the position of F:
Since DF consists of 3 parts and each part is 3.5 units long, the position of F is:
[tex]\[
\text{Position of F} = (-6) + 3 \times 3.5
\][/tex]
Calculate this step-by-step:
[tex]\[
3 \times 3.5 = 10.5
\][/tex]
[tex]\[
(-6) + 10.5 = 4.5
\][/tex]
So, point F is at 4.5 on the number line.
Therefore, the correct answer from the drop-down menu is:
Point [tex]$F$[/tex] is at [tex]\(\boxed{4.5}\)[/tex] on the number line.
