Question

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

Answer 1

To rotate a point \( Q(x, y) \) about the origin by an angle \( \theta \), you can use the rotation formulas:

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

In this case, the point \( Q(-5, -9) \) is rotated by \( -270^\circ \).

First, let's convert the angle to radians for easier calculations. However, we can also directly use the values of sine and cosine corresponding to this angle, as follows:

- For \( -270^\circ \):
- \( \cos(-270^\circ) = 0 \)
- \( \sin(-270^\circ) = -1 \)

Now, applying the formulas with \( x = -5 \) and \( y = -9 \):

\[
x' = -5 \cdot 0 - (-9) \cdot (-1) = 0 - 9 = -9
\]
\[
y' = -5 \cdot (-1) + (-9) \cdot 0 = 5 + 0 = 5
\]

Therefore, the new coordinates after the rotation are:

\[
Q' = (-9, 5)
\]

Thus, the location of point \( Q' \) after rotating \( Q(-5, -9) \) by -270 degrees about the origin is:

\[
\boxed{(-9, 5)}
\]