Subjects geometry

Polar Point Matching 46C88B

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Polar Point Matching 46C88B


1. The problem asks to match each polar coordinate point $(r, \theta)$ with one of the labeled points A, B, C, or D on the graph. 2. Recall that in polar coordinates, a negative radius $r$ means the point is in the opposite direction of the angle $\theta$. To find the equivalent positive radius and angle, add $\pi$ to $\theta$ and take the absolute value of $r$. 3. The points A, B, C, and D correspond to angles $\pi/6$, $5\pi/6$, $7\pi/6$, and $11\pi/6$ respectively, all with radius 2. 4. Now, match each point: - (2, $-11\pi/6$): $-11\pi/6$ is coterminal with $\pi/6$ (since $-11\pi/6 + 2\pi = \pi/6$), radius 2, so matches A. - (-2, $-\pi/6$): Negative radius, so add $\pi$ to angle: $-\pi/6 + \pi = 5\pi/6$, radius 2, matches B. - (-2, $\pi/6$): Negative radius, add $\pi$: $\pi/6 + \pi = 7\pi/6$, radius 2, matches C. - (2, $7\pi/6$): Angle $7\pi/6$ with radius 2, matches C. - (2, $5\pi/6$): Angle $5\pi/6$ with radius 2, matches B. - (-2, $5\pi/6$): Negative radius, add $\pi$: $5\pi/6 + \pi = 11\pi/6$, radius 2, matches D. - (-2, $7\pi/6$): Negative radius, add $\pi$: $7\pi/6 + \pi = 13\pi/6$ which is coterminal with $\pi/6$, radius 2, matches A. - (2, $11\pi/6$): Angle $11\pi/6$ with radius 2, matches D. Final matches: 9. A 10. B 11. C 12. C 13. B 14. D 15. A 16. D