Question

The point Q(−5,−9) is rotated −270° about the origin. What is the location of Q′.

Answer 1

To find the location of the point \( Q' \) after rotating the point \( Q(-5, -9) \) by \( -270^\circ \) about the origin, we can follow these steps:

1. **Understand the rotation direction**: A rotation of \( -270^\circ \) is equivalent to a clockwise rotation of \( 270^\circ \) or a counterclockwise rotation of \( 90^\circ \).

2. **Use the rotation matrix**: The rotation matrix for a counterclockwise rotation by an angle \( \theta \) is given by:
\[
R(\theta) = \begin{bmatrix}
\cos(\theta) & -\sin(\theta) \\
\sin(\theta) & \cos(\theta)
\end{bmatrix}
\]
For \( \theta = 90^\circ \):
\[
R(90^\circ) = \begin{bmatrix}
0 & -1 \\
1 & 0
\end{bmatrix}
\]

3. **Apply the rotation matrix**: We can apply this matrix to the coordinates of point \( Q(-5, -9) \):
\[
Q' = R(90^\circ) \cdot \begin{bmatrix}
-5 \\
-9
\end{bmatrix}
\]
Calculating this gives:
\[
Q' = \begin{bmatrix}
0 & -1 \\
1 & 0
\end{bmatrix} \cdot \begin{bmatrix}
-5 \\
-9
\end{bmatrix} = \begin{bmatrix}
0 \cdot (-5) + (-1) \cdot (-9) \\
1 \cdot (-5) + 0 \cdot (-9)
\end{bmatrix} = \begin{bmatrix}
9 \\
-5
\end{bmatrix}
\]

4. **Locate \( Q' \)**: Therefore, the coordinates of the point \( Q' \) after the rotation are:
\[
Q' = (9, -5)
\]

Thus, the location of the point \( Q' \) after rotating \( Q(-5, -9) \) by \( -270^\circ \) about the origin is \( \boxed{(9, -5)} \).