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Geometry and Trigonometry Difficulty: Medium

The three points shown define a circle. The circumference of this circle is $k π$ $k π$ , where $k$ $k$ is a constant. What is the value of $k$ $k$ ?

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Explanation

Choice B is correct. It’s given that the three points shown define a circle, so the center of that circle is an equal distance from each of the three points. The point $(7, 8)$ $left parenthesis 7 comma 8 right parenthesis$ is halfway between the points $(7, 5)$ $left parenthesis 7 comma 5 right parenthesis$ and $(7, 11)$ $left parenthesis 7 comma 11 right parenthesis$ and is a distance of $3$ $3$ units from each of those two points. The point $(7, 8)$ $left parenthesis 7 comma 8 right parenthesis$ is also a distance of $3$ $3$ units from $(4, 8)$ $left parenthesis 4 comma 8 right parenthesis$ . Because the point $(7, 8)$ $left parenthesis 7 comma 8 right parenthesis$ is the same distance from all three given points, it must be the center of the circle. The radius of a circle is the distance from the center to any point on the circle. Since that distance is $3$ $3$ , it follows that the radius of the circle is $3$ $3$ . The circumference of a circle with radius $r$ $r$ is equal to $2 π r$ $2 pi r$ . It follows that the circumference of the circle is $2 (π) (3)$ $2 left parenthesis pi right parenthesis left parenthesis 3 right parenthesis$ , or $6 π$ $6 pi$ . It's given that the circumference of the circle is $k π$ $k π$ . Therefore, the value of $k$ is $6$ .

Choice A is incorrect. This is the radius of the circle, not the value of $k$ in the expression $k π$ .

Choice C is incorrect. This is the x-coordinate of the center of the circle, not the value of $k$ in the expression $k π$ .

Choice D is incorrect. This is the value of $k$ for which $k π$ represents the area of the circle, in square units, not the circumference of the circle, in units.