rotate (–4,0), (–4,4), (–2, 6), (0,4), and (0,0) 180 degrees counterclockwise.

Answer 1

To rotate the points counterclockwise by 180 degrees, we can use the transformation matrix for a 180 degree counterclockwise rotation:

\[
\begin{bmatrix}
-1 & 0 \\
0 & -1 \\
\end{bmatrix}
\]

Let's apply the rotation to each point:

1. (-4, 0):
\[
\begin{bmatrix}
-1 & 0 \\
0 & -1 \\
\end{bmatrix}
\begin{bmatrix}
-4 \\
0 \\
\end{bmatrix}
=
\begin{bmatrix}
4 \\
0 \\
\end{bmatrix}
\]

So, the rotated coordinates for (-4, 0) are (4, 0).

2. (-4, 4):
\[
\begin{bmatrix}
-1 & 0 \\
0 & -1 \\
\end{bmatrix}
\begin{bmatrix}
-4 \\
4 \\
\end{bmatrix}
=
\begin{bmatrix}
4 \\
-4 \\
\end{bmatrix}
\]

So, the rotated coordinates for (-4, 4) are (4, -4).

3. (-2, 6):
\[
\begin{bmatrix}
-1 & 0 \\
0 & -1 \\
\end{bmatrix}
\begin{bmatrix}
-2 \\
6 \\
\end{bmatrix}
=
\begin{bmatrix}
2 \\
-6 \\
\end{bmatrix}
\]

So, the rotated coordinates for (-2, 6) are (2, -6).

4. (0, 4):
\[
\begin{bmatrix}
-1 & 0 \\
0 & -1 \\
\end{bmatrix}
\begin{bmatrix}
0 \\
4 \\
\end{bmatrix}
=
\begin{bmatrix}
0 \\
-4 \\
\end{bmatrix}
\]

So, the rotated coordinates for (0, 4) are (0, -4).

5. (0, 0):
\[
\begin{bmatrix}
-1 & 0 \\
0 & -1 \\
\end{bmatrix}
\begin{bmatrix}
0 \\
0 \\
\end{bmatrix}
=
\begin{bmatrix}
0 \\
0 \\
\end{bmatrix}
\]

So, the rotated coordinates for (0, 0) are (0, 0).

Therefore, the rotated points are: (4, 0), (4, -4), (2, -6), (0, -4), and (0, 0).