Determine the equation of the circle graphed below. 12 11 10 9. 8 (7, 0) 1 2 3 4 56 789 10 11 12 -12-11-10-9 -8 -7 -6 -5 -4 -3 -2 -1, 3 (4, -4). 15 -7 -8 -9 -10 -11 -12 76 543 21
Determine the equation of the circle graphed below. 12 11 10 9. 8 (7, 0) 1 2 3 4 56 789 10 11 12 -12-11-10-9 -8 -7 -6 -5 -4 -3 -2 -1, 3 (4, -4). 15 -7 -8 -9 -10 -11 -12 76 543 21
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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![**Title: Determine the Equation of the Circle**
**Objective:**
Learn how to determine the equation of a circle given its graph.
**Graph Description:**
The graph depicts a circle on a Cartesian plane with labeled axes.
- **Axes:**
- The horizontal axis (x-axis) and vertical axis (y-axis) both range from -12 to 12.
- The origin (0,0) is centered where the x-axis and y-axis intersect.
- **Circle Details:**
- The circle is centered at the point (4, -4), as indicated on the graph.
- A point on the circumference of the circle is marked at (7,0).
**Points:**
- Center: (4, -4)
- Edge Point: (7, 0)
**Finding the Radius:**
The radius can be found using the distance formula between the center of the circle (4, -4) and any point on the circumference (7, 0). The distance formula is:
\[ \text{Radius} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
\[ = \sqrt{(7 - 4)^2 + (0 + 4)^2} \]
\[ = \sqrt{3^2 + 4^2} \]
\[ = \sqrt{9 + 16} \]
\[ = \sqrt{25} \]
\[ = 5 \]
**Equation of the Circle:**
The standard form of a circle's equation with center \((h, k)\) and radius \(r\) is:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Given:
- Center, \((h, k)\) = (4, -4)
- Radius, \(r\) = 5
Hence, the equation is:
\[ (x - 4)^2 + (y + 4)^2 = 5^2 \]
\[ (x - 4)^2 + (y + 4)^2 = 25 \]
**Conclusion:**
The equation of the circle represented in the graph is \((x - 4)^2 + (y + 4)^2 = 25\).
Make sure to practice identifying the center and radius of circles on graphs to become proficient in deriving their equations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa2f2b4b7-5c6b-401d-b172-513b58592158%2Fbae86f5b-ca4e-4bc5-803e-1f946b3e15d7%2Fbbjtu5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Determine the Equation of the Circle**
**Objective:**
Learn how to determine the equation of a circle given its graph.
**Graph Description:**
The graph depicts a circle on a Cartesian plane with labeled axes.
- **Axes:**
- The horizontal axis (x-axis) and vertical axis (y-axis) both range from -12 to 12.
- The origin (0,0) is centered where the x-axis and y-axis intersect.
- **Circle Details:**
- The circle is centered at the point (4, -4), as indicated on the graph.
- A point on the circumference of the circle is marked at (7,0).
**Points:**
- Center: (4, -4)
- Edge Point: (7, 0)
**Finding the Radius:**
The radius can be found using the distance formula between the center of the circle (4, -4) and any point on the circumference (7, 0). The distance formula is:
\[ \text{Radius} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
\[ = \sqrt{(7 - 4)^2 + (0 + 4)^2} \]
\[ = \sqrt{3^2 + 4^2} \]
\[ = \sqrt{9 + 16} \]
\[ = \sqrt{25} \]
\[ = 5 \]
**Equation of the Circle:**
The standard form of a circle's equation with center \((h, k)\) and radius \(r\) is:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Given:
- Center, \((h, k)\) = (4, -4)
- Radius, \(r\) = 5
Hence, the equation is:
\[ (x - 4)^2 + (y + 4)^2 = 5^2 \]
\[ (x - 4)^2 + (y + 4)^2 = 25 \]
**Conclusion:**
The equation of the circle represented in the graph is \((x - 4)^2 + (y + 4)^2 = 25\).
Make sure to practice identifying the center and radius of circles on graphs to become proficient in deriving their equations.
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