To find the center of a circle given the endpoints of its diameter, we need to determine the midpoint of the line segment connecting these two endpoints. The midpoint formula for a line segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
In this problem, the endpoints of the diameter are [tex]\((7,4)\)[/tex] and [tex]\((-1,6)\)[/tex]. Let's calculate the midpoint step by step:
1. Calculate the x-coordinate of the midpoint:
[tex]\[
\frac{x_1 + x_2}{2} = \frac{7 + (-1)}{2} = \frac{7 - 1}{2} = \frac{6}{2} = 3.0
\][/tex]
2. Calculate the y-coordinate of the midpoint:
[tex]\[
\frac{y_1 + y_2}{2} = \frac{4 + 6}{2} = \frac{10}{2} = 5.0
\][/tex]
Therefore, the coordinates of the center point of the circle are [tex]\((3.0, 5.0)\)[/tex].
Given the options:
- (-1,4)
- (7,6)
- (6,10)
- (3,5)
The correct answer is [tex]\((3,5)\)[/tex].
