To find the radius of a circle given the endpoints of its diameter, we can follow these steps: ### Step 1: Understand the Diameter The diameter of a circle is the longest distance across the circle, passing through the center. The endpoints of the diameter are two points on the circle. In this case, the endpoints are given as \( P(-6, 1) \) and \( Q(6, 6) \). ### Step 2: Calculate the Length of the Diameter To find the length of the diameter, we can use the distance formula, which is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Where: - \( (x_1, y_1) \) are the coordinates of point \( P \) - \( (x_2, y_2) \) are the coordinates of point \( Q \) Substituting the coordinates of points \( P \) and \( Q \): - \( P(-6, 1) \) gives \( x_1 = -6 \) and \( y_1 = 1 \) - \( Q(6, 6) \) gives \( x_2 = 6 \) and \( y_2 = 6 \) Now, plug these values into the distance formula: \[ d = \sqrt{(6 - (-6))^2 + (6 - 1)^2} \] ### Step 3: Simplify the Calculation Calculating the differences: - \( 6 - (-6) = 6 + 6 = 12 \) - \( 6 - 1 = 5 \) Now substitute these values back into the formula: \[ d = \sqrt{(12)^2 + (5)^2} \] Calculating the squares: \[ d = \sqrt{144 + 25} \] Adding these together: \[ d = \sqrt{169} \] Taking the square root gives: \[ d = 13 \] ### Step 4: Calculate the Radius The radius \( r \) of the circle is half the length of the diameter: \[ r = \frac{d}{2} = \frac{13}{2} = 6.5 \] ### Conclusion Thus, the radius of the circle is **6.5 units**. ### Explanation of Other Options - **Option A (6.5 units)**: This is the correct answer as we calculated the radius to be 6.5 units. - **Option B (13.0 units)**: This option represents the length of the diameter, not the radius. The radius is half of the diameter. - **Option C (3.5 units)**: This is incorrect as it does not relate to the calculated diameter or radius. It is less than half of the diameter. - **Option D (7.0 units)**: This is also incorrect as it is slightly more than half of the diameter. The correct radius is 6.5 units. ### Revision Summary - The diameter of a circle is the distance between two points on the circle. - Use the distance formula to calculate the length of the diameter. - The radius is half the length of the diameter. - The correct radius for the given points is 6.5 units.