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What is the center point of a circle with a diameter with endpoints at (7, 4) and (-1, 6)?

A. (-1, 4)
B. (7, 6)
C. (6, 10)
D. (3, 5)


Sagot :

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].