Question

point q was rotated about the origin (0, 0) by -165°. which point is the image of q? choose 1 answer: a a b b c c d d

Explanation:

Step1: Identify coordinates of Q

Point Q is at $(-2, 5)$.

Step2: Understand rotation direction

A rotation of $-165^\circ$ is equivalent to a clockwise rotation of $165^\circ$, or a counterclockwise rotation of $360^\circ - 165^\circ = 195^\circ$.

Step3: Apply rotation formula

The rotation formula for a point $(x,y)$ about the origin by $\theta$ is:
$$x' = x\cos\theta - y\sin\theta$$
$$y' = x\sin\theta + y\cos\theta$$
Using $\theta = -165^\circ$:
$\cos(-165^\circ) = \cos(165^\circ) = -\cos(15^\circ) \approx -0.9659$
$\sin(-165^\circ) = -\sin(165^\circ) = -\sin(15^\circ) \approx -0.2588$
Calculate $x'$:
$x' = (-2)(-0.9659) - (5)(-0.2588) = 1.9318 + 1.294 = 3.2258$
Calculate $y'$:
$y' = (-2)(-0.2588) + (5)(-0.9659) = 0.5176 - 4.8295 = -4.3119$

Step4: Match to given points

The calculated coordinates $(3.23, -4.31)$ are closest to point D, which is at $(4, -4)$.

Answer:

D. D