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The endpoints of a diameter of a circle are left parenthesis negative 1 comma 3 right parenthesis and left parenthesis 3 comma 1 right parenthesis. Find the center and radius of the circle.
(1 point)
Responses

center: left parenthesis negative 2 comma 1 right parenthesis radius: start root 5 end root
center: Image with alt text: left parenthesis negative 2 comma 1 right parenthesis radius: Image with alt text: start root 5 end root

center: left parenthesis negative 2 comma 1 right parenthesiss radius: 5
center: Image with alt text: left parenthesis negative 2 comma 1 right parenthesiss radius: Image with alt text: 5

center: left parenthesis 1 comma 2 right parenthesis radius: start root 5 end root
center: Image with alt text: left parenthesis 1 comma 2 right parenthesis radius: Image with alt text: start root 5 end root

center: left parenthesis 1 comma 2 right parenthesis radius: 5

Answer 1

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: \(\sqrt{5}\)