Question

The point Q(-5,-9) is rotated -270 degrees about the origin. Select the location of Q'
A. Q' (5,9)
B. Q' (9,-5
C. Q'(-5,9)
D. Q'(9,5)

Answer 1

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)**.