To find the center and radius of a circle with given endpoints of a diameter, we first determine the center using the midpoint formula.
The endpoints of the diameter are given as \((-1, 3)\) and \((3, 1)\).
Step 1: Find the center (midpoint)
The midpoint \(M\) of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the values:
\[ M = \left( \frac{-1 + 3}{2}, \frac{3 + 1}{2} \right) = \left( \frac{2}{2}, \frac{4}{2} \right) = (1, 2) \]
So, the center of the circle is \((1, 2)\).
Step 2: Find the radius
The radius is half the distance between the endpoints of the diameter. We can use the distance formula to find the distance between the two points.
The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Calculating the distance between the points \((-1, 3)\) and \((3, 1)\):
\[ d = \sqrt{(3 - (-1))^2 + (1 - 3)^2} = \sqrt{(3 + 1)^2 + (1 - 3)^2} = \sqrt{4^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \]
The radius \(r\) is half of the diameter, so:
\[ r = \frac{d}{2} = \frac{2\sqrt{5}}{2} = \sqrt{5} \]
Final Result:
Center: \((1, 2)\)
Radius: \(\sqrt{5}\)
Thus, the correct response is:
center: \((1, 2)\) radius: \(\sqrt{5}\)
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