To rotate a point \((x, y)\) about the origin by an angle \(\theta\), you can use the rotation formulas:
\[ x' = x \cos(\theta) - y \sin(\theta) \] \[ y' = x \sin(\theta) + y \cos(\theta) \]
In this case, the point \(Q(-5, -9)\) is being rotated by \(-270\) degrees.
First, we convert \(-270\) degrees to radians (though it's not necessary for our cosine and sine):
\[ -270^\circ = -\frac{3\pi}{2} \text{ radians} \]
Next, we determine the values for \(\cos(-270^\circ)\) and \(\sin(-270^\circ)\):
\[ \cos(-270^\circ) = 0 \] \[ \sin(-270^\circ) = 1 \]
Now we substitute these values into our rotation formulas:
\[ x' = -5 \cdot 0 - (-9) \cdot 1 = 0 + 9 = 9 \] \[ y' = -5 \cdot 1 + (-9) \cdot 0 = -5 + 0 = -5 \]
So the new coordinates for \(Q'\) are:
\[ Q' (9, -5) \]
Therefore, the answer is B. Q' (9, -5).
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