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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Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
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A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
Question
![### Problem Statement:
Given three points \(A\), \(B\), and \(C\) on a line segment, we know the following:
- \(AC = 16\)
- \(AB = x + 1\)
- \(BC = x + 7\)
The problem is to determine the measure of the length of \( \overline{AB} \).
### Diagram:
The diagram shows a straight line segment with:
- Point \(A\) at the left end.
- Point \(B\) somewhere between points \(A\) and \(C\) (closer to \(A\)).
- Point \(C\) at the right end.
Below is the diagram:
\[ A \quad \quad \quad \quad B \quad \quad \quad \quad C \]
\[ \bullet \quad \quad \quad \bullet \quad \quad \quad \bullet \]
\[ \quad x + 1 \quad \quad \quad \quad x + 7 \]
### Equations and Solution:
We know from the problem statement that:
\[ AB + BC = AC \]
Substituting the given lengths:
\[ (x + 1) + (x + 7) = 16 \]
\[ x + 1 + x + 7 = 16 \]
\[ 2x + 8 = 16 \]
Subtracting 8 from both sides:
\[ 2x = 8 \]
Dividing by 2:
\[ x = 4 \]
Now, substituting \(x = 4\) back into \(AB\):
\[ AB = x + 1 = 4 + 1 = 5 \]
### Answer:
\[ \boxed{5} \]
Multiple choice options:
- \(\bullet\) 4
- \( \circ\) 5
- \( \circ\) 11
- \( \circ\) 16
The correct answer is **5**. Thus, the measure of the length of \( \overline{AB} \) is 5 units.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa4ec90bd-5b1a-4ba1-9571-0f30153ca54b%2F06742691-3d06-46bc-82fa-3ee36bc49a14%2Fy8huup_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement:
Given three points \(A\), \(B\), and \(C\) on a line segment, we know the following:
- \(AC = 16\)
- \(AB = x + 1\)
- \(BC = x + 7\)
The problem is to determine the measure of the length of \( \overline{AB} \).
### Diagram:
The diagram shows a straight line segment with:
- Point \(A\) at the left end.
- Point \(B\) somewhere between points \(A\) and \(C\) (closer to \(A\)).
- Point \(C\) at the right end.
Below is the diagram:
\[ A \quad \quad \quad \quad B \quad \quad \quad \quad C \]
\[ \bullet \quad \quad \quad \bullet \quad \quad \quad \bullet \]
\[ \quad x + 1 \quad \quad \quad \quad x + 7 \]
### Equations and Solution:
We know from the problem statement that:
\[ AB + BC = AC \]
Substituting the given lengths:
\[ (x + 1) + (x + 7) = 16 \]
\[ x + 1 + x + 7 = 16 \]
\[ 2x + 8 = 16 \]
Subtracting 8 from both sides:
\[ 2x = 8 \]
Dividing by 2:
\[ x = 4 \]
Now, substituting \(x = 4\) back into \(AB\):
\[ AB = x + 1 = 4 + 1 = 5 \]
### Answer:
\[ \boxed{5} \]
Multiple choice options:
- \(\bullet\) 4
- \( \circ\) 5
- \( \circ\) 11
- \( \circ\) 16
The correct answer is **5**. Thus, the measure of the length of \( \overline{AB} \) is 5 units.
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