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Difficulty: Hard

Point $F$ lies on a unit circle in the xy-plane and has coordinates $left parenthesis 1 comma 0 right parenthesis$ . Point $G$ is the center of the circle and has coordinates $left parenthesis 0 comma 0 right parenthesis$ . Point $H$ also lies on the circle and has coordinates $left parenthesis negative 1 comma y right parenthesis$ , where $y$ is a constant. Which of the following could be the positive measure of angle $F G H$ , in radians?

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Explanation

Choice D is correct. It's given that the circle is a unit circle, which means the circle has a radius of $1$ . It's also given that point $G$ is the center of the circle and has coordinates $left parenthesis 0 comma 0 right parenthesis$ and that point $H$ lies on the circle and has coordinates $left parenthesis negative 1 comma y right parenthesis$ . Since the radius of the circle is $1$ , the value of $y$ must be $0$ , as all other points with an x-coordinate of $negative 1$ are a distance greater than $1$ away from point $G$ . Since $F$ and $H$ are points on the unit circle centered at $G$ , let side $F G$ be the starting side of the angle and side $G H$ be the terminal side of the angle. Since side $F G$ is on the positive x-axis and side $G H$ is on the negative x-axis, side $F G$ is half of a rotation around the unit circle, or $π$ radians, away from side $G H$ . Therefore, the positive measure of angle $F G H$ , in radians, is equal to $π$ plus an integer multiple of $2 pi$ . In other words, the positive measure of angle $F G H$ , in radians, is an odd integer multiple of $π$ . Of the given choices, only $25 pi$ is an odd integer multiple of $π$ .

Choice A is incorrect. This could be the positive measure of an angle where the starting side is $F G$ and the terminal side contains the point $left parenthesis 0 comma negative 1 right parenthesis$ , not $left parenthesis negative 1 comma 0 right parenthesis$ .

Choice B is incorrect. This could be the positive measure of an angle where the starting side is $F G$ and the terminal side contains the point $left parenthesis 0 comma 1 right parenthesis$ , not $left parenthesis negative 1 comma 0 right parenthesis$ .

Choice C is incorrect. This could be the positive measure of an angle where the starting side is $F G$ and the terminal side contains the point $left parenthesis 1 comma 0 right parenthesis$ , not $left parenthesis negative 1 comma 0 right parenthesis$ .