Let's solve this problem step-by-step:
1. Identify Given:
- Point [tex]\( A \)[/tex] is located at 6.
- Point [tex]\( C \)[/tex] is located between [tex]\( A \)[/tex] and [tex]\( B \)[/tex] at 1.875.
- The ratio [tex]\( AC : CB = 3 : 5 \)[/tex].
2. Understand the Ratio:
- If we let the distance from [tex]\( C \)[/tex] to [tex]\( B \)[/tex] be [tex]\( x \)[/tex], then the distance from [tex]\( A \)[/tex] to [tex]\( C \)[/tex] can be expressed using the given ratio.
- Since [tex]\( AC : CB = 3 : 5 \)[/tex], we can write [tex]\( AC = 3x \)[/tex] and [tex]\( CB = 5x \)[/tex].
3. Find Distance [tex]\( AC \)[/tex]:
- The distance from [tex]\( A \)[/tex] to [tex]\( C \)[/tex] is the difference between their coordinates:
[tex]\[
AC = 6 - 1.875 = 4.125 \text{ units}
\][/tex]
4. Use the Ratio to Find [tex]\( CB \)[/tex]:
- Since [tex]\( AC = 3x \)[/tex], we have:
[tex]\[
4.125 = 3x
\][/tex]
- Solving for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{4.125}{3}
\][/tex]
5. Calculate [tex]\( CB \)[/tex]:
- Because [tex]\( CB = 5x \)[/tex]:
[tex]\[
CB = 5 \times \frac{4.125}{3} = \frac{5 \times 4.125}{3} = 6.875 \text{ units}
\][/tex]
6. Find the Total Distance [tex]\( AB \)[/tex]:
- The total distance [tex]\( AB \)[/tex] is the sum of distances [tex]\( AC \)[/tex] and [tex]\( CB \)[/tex]:
[tex]\[
AB = AC + CB = 4.125 + 6.875 = 11 \text{ units}
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{11}
\][/tex]
So, the length of [tex]\( \overline{AB} \)[/tex] is [tex]\( 11 \)[/tex] units, which corresponds to option (C) [tex]\( AB = 11 \)[/tex] units.
