Question

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

Answer 1

To rotate a point \((-5, -9)\) around the origin, we can use the rotation transformation formula. Rotating a point \((x, y)\) by an angle \(\theta\) (in degrees) around the origin can be expressed with the following formulas:

\[
x' = x \cos(\theta) - y \sin(\theta)
\]
\[
y' = x \sin(\theta) + y \cos(\theta)
\]

For \(-270^\circ\), we can express it in terms of a positive angle:
\[
-270^\circ = 90^\circ
\]

So, we need to use \(\theta = 90^\circ\):
\[
\cos(90^\circ) = 0, \quad \sin(90^\circ) = 1
\]

Now, applying the formulas to the point \(Q(-5, -9)\):

1. Calculate \(x'\):
\[
x' = -5 \cdot 0 - (-9) \cdot 1 = 0 + 9 = 9
\]

2. Calculate \(y'\):
\[
y' = -5 \cdot 1 + (-9) \cdot 0 = -5 + 0 = -5
\]

Thus, the coordinates after the rotation are:
\[
Q' = (9, -5)
\]

Therefore, the correct answer is:
**C. \(Q' (9, -5)\)**.