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
Related questions
Question
Example 8
![**Example 8:**
Given that \( \overline{AB} \) and \( \overline{BC} \) are tangent to circle \( P \).
If \( AB = 2x - 7 \) and \( CB = 5x - 52 \), find the value of \( x \) and the length of \( \overline{AB} \).
### Solution:
Since \( AB = BC \):
1. Set the equations equal to each other:
\[
3x - 24 = x + 50
\]
2. Simplify the equation:
\[
2x - 24 = 50
\]
3. Solve for \( x \):
\[
2x = 74
\]
\[
x = 37
\]
### Diagrams:
There are two diagrams showing circle \( P \), tangents \( AB \) and \( BC \), and their respective points of tangency:
1. The first diagram shows the points \( B \) and \( C \) on the circle, with lines \( 3x - 24 \) and \( x + 50 \) respectively denoting the tangential distances.
2. The second diagram visually represents the geometric setup where the tangents meet the circle.
These diagrams help illustrate the problem by showing the relationship between the tangents and the circle they touch.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F88296449-4020-45ce-a35f-28e4157e48b0%2Fb474cad7-ef6a-497c-a36f-9d5ccacdd722%2Fznqa4mo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Example 8:**
Given that \( \overline{AB} \) and \( \overline{BC} \) are tangent to circle \( P \).
If \( AB = 2x - 7 \) and \( CB = 5x - 52 \), find the value of \( x \) and the length of \( \overline{AB} \).
### Solution:
Since \( AB = BC \):
1. Set the equations equal to each other:
\[
3x - 24 = x + 50
\]
2. Simplify the equation:
\[
2x - 24 = 50
\]
3. Solve for \( x \):
\[
2x = 74
\]
\[
x = 37
\]
### Diagrams:
There are two diagrams showing circle \( P \), tangents \( AB \) and \( BC \), and their respective points of tangency:
1. The first diagram shows the points \( B \) and \( C \) on the circle, with lines \( 3x - 24 \) and \( x + 50 \) respectively denoting the tangential distances.
2. The second diagram visually represents the geometric setup where the tangents meet the circle.
These diagrams help illustrate the problem by showing the relationship between the tangents and the circle they touch.
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