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Select the correct answer.

Point A lies on a number line at 6. Point [tex]C[/tex] lies between points [tex]B[/tex] and [tex]A[/tex] at 1.875. The ratio [tex]AC: CB = 3: 5[/tex]. What is the length of [tex]\overline{AB}[/tex]?

A. [tex]AB = 15[/tex] units
B. [tex]AB = 8.75[/tex] units
C. [tex]AB = 11[/tex] units
D. [tex]AB = 6.875[/tex] units


Sagot :

Let's work through the problem step-by-step.

1. Identify the given information:
- Point A is at 6.
- Point C is at 1.875.
- The ratio \( AC:CB = 3:5 \).

2. Determine the length of \( AC \):
- Since point A is at 6 and point C is at 1.875, the distance \( AC \) can be found by subtracting the coordinate of point C from point A:
[tex]\[ AC = 6 - 1.875 = 4.125 \text{ units} \][/tex]

3. Set up the ratio equation:
- \( AC \) and \( CB \) are in the ratio of \( 3:5 \). Let the length of \( CB \) be \( x \) units.
- Then, \( AC = \frac{3}{5} x \).

4. Solve for \( x \):
- We already found that \( AC \) is 4.125 units. Using the ratio equation, we get:
[tex]\[ 4.125 = \frac{3}{5} x \][/tex]
- To solve for \( x \), multiply both sides by \( \frac{5}{3} \):
[tex]\[ x = 4.125 \times \frac{5}{3} = 6.875 \text{ units} \][/tex]
- Thus, \( CB = 6.875 \text{ units} \).

5. Find the total length \( AB \):
- \( AB = AC + CB \).
- So,
[tex]\[ AB = 4.125 + 6.875 = 11.0 \text{ units} \][/tex]

Therefore, the length of \( \overline{AB} \) is \( 11 \) units, and the correct answer is:

C. [tex]\( AB = 11 \)[/tex] units.