Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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Question
![### Problem Description
In the given image, we have a line segment with points A, B, and C. The distances between these points are labeled as follows:
- \(AB = x + 1\)
- \(BC = x + 7\)
The total length of \(AC\) is given by:
- \(AC = 16\)
The problem is to find the measure of the length of segment \(\overline{AB}\).
### Provided Options
1. 4
2. 5
3. 11
4. 16
### Additional Diagram Description
The diagram shows a horizontal line with three points labeled:
- Point A
- Point B, which is at a distance of \(x + 1\) from point A
- Point C, which is at a distance of \(x + 7\) from point B
### Solution Explanation
To solve for the length of \(\overline{AB}\), we use the fact that the sum of segments \(AB\) and \(BC\) equals the total length \(AC\):
\[
AB + BC = AC
\]
Substitute the given expressions for \(AB\), \(BC\), and \(AC\):
\[
(x + 1) + (x + 7) = 16
\]
Combine like terms:
\[
x + 1 + x + 7 = 16
\]
\[
2x + 8 = 16
\]
To solve for \(x\), subtract 8 from both sides:
\[
2x = 16 - 8
\]
\[
2x = 8
\]
Divide by 2:
\[
x = 4
\]
Now, substitute \(x\) back into the expression for \(AB\):
\[
AB = x + 1 = 4 + 1 = 5
\]
Thus, the length of \(\overline{AB}\) is **5**.
### Answer
The measure of the length of \(\overline{AB}\) is \( \boxed{5} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5e777900-48b9-4d96-bed3-a41476812193%2Fc8bc5176-405d-439f-b0bf-a85b009be564%2Fxqsuw3v_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Description
In the given image, we have a line segment with points A, B, and C. The distances between these points are labeled as follows:
- \(AB = x + 1\)
- \(BC = x + 7\)
The total length of \(AC\) is given by:
- \(AC = 16\)
The problem is to find the measure of the length of segment \(\overline{AB}\).
### Provided Options
1. 4
2. 5
3. 11
4. 16
### Additional Diagram Description
The diagram shows a horizontal line with three points labeled:
- Point A
- Point B, which is at a distance of \(x + 1\) from point A
- Point C, which is at a distance of \(x + 7\) from point B
### Solution Explanation
To solve for the length of \(\overline{AB}\), we use the fact that the sum of segments \(AB\) and \(BC\) equals the total length \(AC\):
\[
AB + BC = AC
\]
Substitute the given expressions for \(AB\), \(BC\), and \(AC\):
\[
(x + 1) + (x + 7) = 16
\]
Combine like terms:
\[
x + 1 + x + 7 = 16
\]
\[
2x + 8 = 16
\]
To solve for \(x\), subtract 8 from both sides:
\[
2x = 16 - 8
\]
\[
2x = 8
\]
Divide by 2:
\[
x = 4
\]
Now, substitute \(x\) back into the expression for \(AB\):
\[
AB = x + 1 = 4 + 1 = 5
\]
Thus, the length of \(\overline{AB}\) is **5**.
### Answer
The measure of the length of \(\overline{AB}\) is \( \boxed{5} \).
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