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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![### Analytical Geometry Problems
#### Question 20
**Find the distance between the given points.**
Enter square roots using "sqrt" or round to the nearest 10th.
Points: (5, 1) and (-2, 6)
_To solve this problem, use the distance formula:_
\[ \text{Distance} = \sqrt{{(x_2 - x_1)^2 + (y_2 - y_1)^2}} \]
1. Plug in the coordinates of the given points:
\[ (x_1, y_1) = (5, 1) \]
\[ (x_2, y_2) = (-2, 6) \]
2. Substitute the values into the distance formula:
\[ \text{Distance} = \sqrt{{(-2 - 5)^2 + (6 - 1)^2}} \]
3. Perform the calculations within the square root:
\[ \text{Distance} = \sqrt{{(-7)^2 + (5)^2}} \]
\[ \text{Distance} = \sqrt{{49 + 25}} \]
\[ \text{Distance} = \sqrt{74} \]
4. To express the square root using "sqrt":
\[ \text{Distance} = \sqrt{74} \]
5. To round to the nearest 10th, approximate the square root:
\[ \sqrt{74} \approx 8.6 \]
Therefore, the distance between the points (5, 1) and (-2, 6) is either \( \sqrt{74} \) or approximately 8.6.
#### Question 21
**Write the standard form of a circle with a center at C(-7, 8) and passing through the point (1, 3).**
_To solve this problem, use the standard form of a circle's equation and the distance formula:_
\[ (x - h)^2 + (y - k)^2 = r^2 \]
1. Identify the center of the circle \((h, k)\):
\[ (h, k) = (-7, 8) \]
2. Use the distance formula to find the radius \( r \), the distance from the center to the point (1, 3):
\[ r = \sqrt{{(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa7c29d2f-5cf3-43f5-aa90-abc5a822140a%2Fb4b7dd35-a3a6-484f-ab59-1843bee78318%2F28j5k2q_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Analytical Geometry Problems
#### Question 20
**Find the distance between the given points.**
Enter square roots using "sqrt" or round to the nearest 10th.
Points: (5, 1) and (-2, 6)
_To solve this problem, use the distance formula:_
\[ \text{Distance} = \sqrt{{(x_2 - x_1)^2 + (y_2 - y_1)^2}} \]
1. Plug in the coordinates of the given points:
\[ (x_1, y_1) = (5, 1) \]
\[ (x_2, y_2) = (-2, 6) \]
2. Substitute the values into the distance formula:
\[ \text{Distance} = \sqrt{{(-2 - 5)^2 + (6 - 1)^2}} \]
3. Perform the calculations within the square root:
\[ \text{Distance} = \sqrt{{(-7)^2 + (5)^2}} \]
\[ \text{Distance} = \sqrt{{49 + 25}} \]
\[ \text{Distance} = \sqrt{74} \]
4. To express the square root using "sqrt":
\[ \text{Distance} = \sqrt{74} \]
5. To round to the nearest 10th, approximate the square root:
\[ \sqrt{74} \approx 8.6 \]
Therefore, the distance between the points (5, 1) and (-2, 6) is either \( \sqrt{74} \) or approximately 8.6.
#### Question 21
**Write the standard form of a circle with a center at C(-7, 8) and passing through the point (1, 3).**
_To solve this problem, use the standard form of a circle's equation and the distance formula:_
\[ (x - h)^2 + (y - k)^2 = r^2 \]
1. Identify the center of the circle \((h, k)\):
\[ (h, k) = (-7, 8) \]
2. Use the distance formula to find the radius \( r \), the distance from the center to the point (1, 3):
\[ r = \sqrt{{(
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