Question:
The point with coordinates \((-6,-7)\) is the center of circle \( C \). The point \((-6,5)\) lies inside circle \( C \), and the point \((8,-7)\) lies outside circle \( C \). What is the radius of circle \( C \)?
The point with coordinates \((-6,-7)\) is the center of circle \( C \). The point \((-6,5)\) lies inside circle \( C \), and the point \((8,-7)\) lies outside circle \( C \). What is the radius of circle \( C \)?
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When dealing with circles and coordinates, use the distance formula between the center and other points to establish bounds for the radius.
Updated On: Oct 7, 2025
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The Correct Option is D
Solution and Explanation
Step 1: Distance from center to point inside.
The distance between \((-6,-7)\) and \((-6,5)\) is: \[ |5 - (-7)| = 12. \] Thus, the radius must be greater than 12.
Step 2: Distance from center to point outside.
The distance between \((-6,-7)\) and \((8,-7)\) is: \[ |8 - (-6)| = 14. \] Thus, the radius must be less than 14.
Step 3: Integer restriction.
Since the radius is an integer, it must be 13.
Step 4: Conclusion.
The radius of the circle is: \[ \boxed{\text{(D) 13}} \]
The distance between \((-6,-7)\) and \((-6,5)\) is: \[ |5 - (-7)| = 12. \] Thus, the radius must be greater than 12.
Step 2: Distance from center to point outside.
The distance between \((-6,-7)\) and \((8,-7)\) is: \[ |8 - (-6)| = 14. \] Thus, the radius must be less than 14.
Step 3: Integer restriction.
Since the radius is an integer, it must be 13.
Step 4: Conclusion.
The radius of the circle is: \[ \boxed{\text{(D) 13}} \]
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