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What is the center of the circle that has a diameter whose endpoints are [tex]\((2, 7)\)[/tex] and [tex]\((-6, -1)\)[/tex]?

Sagot :

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].
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