To find the center of a circle given the endpoints of its diameter, we need to determine the midpoint of the segment connecting these two endpoints.
Let's denote the endpoints of the diameter as [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex]. Specifically, we have:
[tex]\[ (x_1, y_1) = (2, 7) \][/tex]
[tex]\[ (x_2, y_2) = (-6, -1) \][/tex]
The formula for the midpoint [tex]\((x_m, y_m)\)[/tex] of a segment connecting points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ x_m = \frac{x_1 + x_2}{2} \][/tex]
[tex]\[ y_m = \frac{y_1 + y_2}{2} \][/tex]
First, we compute the x-coordinate of the midpoint:
[tex]\[ x_m = \frac{2 + (-6)}{2} \][/tex]
[tex]\[ x_m = \frac{2 - 6}{2} \][/tex]
[tex]\[ x_m = \frac{-4}{2} \][/tex]
[tex]\[ x_m = -2 \][/tex]
Next, we compute the y-coordinate of the midpoint:
[tex]\[ y_m = \frac{7 + (-1)}{2} \][/tex]
[tex]\[ y_m = \frac{7 - 1}{2} \][/tex]
[tex]\[ y_m = \frac{6}{2} \][/tex]
[tex]\[ y_m = 3 \][/tex]
Therefore, the center of the circle, which is the midpoint of the diameter, is:
[tex]\[ (-2, 3) \][/tex]
This point [tex]\((-2, 3)\)[/tex] is the center of the circle that has a diameter with endpoints [tex]\((2, 7)\)[/tex] and [tex]\((-6, -1)\)[/tex].
