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
![**Question 11: Prove or Disprove the Statement**
The statement to evaluate is: "The point (2, 3) lies on the circle centered at the origin with a radius of 8."
### Solution:
To determine if the point (2, 3) lies on the circle centered at the origin with radius 8, we use the circle equation:
\[ x^2 + y^2 = r^2 \]
where \( (x, y) \) is any point on the circle, and \( r \) is the radius.
1. Substitute the given point \((2, 3)\) and the radius \( r = 8 \) into the equation:
\[ 2^2 + 3^2 = 8^2 \]
2. Calculate the left-hand side of the equation:
\[ 2^2 + 3^2 = 4 + 9 = 13 \]
3. Calculate the right-hand side of the equation:
\[ 8^2 = 64 \]
Since \( 13 \neq 64 \), the point \( (2, 3) \) does not satisfy the equation of the circle centered at the origin with a radius of 8.
### Conclusion:
The statement is disproved. The point (2, 3) does not lie on the circle centered at the origin with radius 8.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F42779bdc-66e8-47e8-8a59-0cd69b39b8c3%2Fee69c8b5-3620-4e93-869e-149ee8c0d6af%2Fyjvyu9d_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question 11: Prove or Disprove the Statement**
The statement to evaluate is: "The point (2, 3) lies on the circle centered at the origin with a radius of 8."
### Solution:
To determine if the point (2, 3) lies on the circle centered at the origin with radius 8, we use the circle equation:
\[ x^2 + y^2 = r^2 \]
where \( (x, y) \) is any point on the circle, and \( r \) is the radius.
1. Substitute the given point \((2, 3)\) and the radius \( r = 8 \) into the equation:
\[ 2^2 + 3^2 = 8^2 \]
2. Calculate the left-hand side of the equation:
\[ 2^2 + 3^2 = 4 + 9 = 13 \]
3. Calculate the right-hand side of the equation:
\[ 8^2 = 64 \]
Since \( 13 \neq 64 \), the point \( (2, 3) \) does not satisfy the equation of the circle centered at the origin with a radius of 8.
### Conclusion:
The statement is disproved. The point (2, 3) does not lie on the circle centered at the origin with radius 8.
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