To rotate the given triangle [tex]\(180^{\circ}\)[/tex] counter-clockwise about the origin, we will follow these steps:
1. Understand the Coordinates of the Triangle:
The original coordinates of the triangle are given as:
[tex]\[
\begin{array}{cc}
\left(\begin{array}{ccc}
0 & -3 & 5 \\
0 & 1 & 2
\end{array}\right)
\end{array}
\][/tex]
Here, the coordinates represent three points [tex]\((0, 0)\)[/tex], [tex]\((-3, 1)\)[/tex], and [tex]\((5, 2)\)[/tex].
2. Rotation Matrix for [tex]\(180^{\circ}\)[/tex]:
The rotation matrix for a [tex]\(180^{\circ}\)[/tex] counter-clockwise rotation about the origin is:
[tex]\[
\left(\begin{array}{cc}
-1 & 0 \\
0 & -1
\end{array}\right)
\][/tex]
3. Apply the Rotation:
To rotate each point, we will multiply the rotation matrix by each coordinate pair.
- For point [tex]\((0, 0)\)[/tex]:
[tex]\[
\left( \begin{array}{cc}
-1 & 0 \\
0 & -1
\end{array} \right)
\left( \begin{array}{c}
0 \\
0
\end{array} \right)
= \left( \begin{array}{c}
0 \\
0
\end{array} \right)
\][/tex]
The rotated coordinates for this point are [tex]\((0, 0)\)[/tex].
- For point [tex]\((-3, 1)\)[/tex]:
[tex]\[
\left( \begin{array}{cc}
-1 & 0 \\
0 & -1
\end{array} \right)
\left( \begin{array}{c}
-3 \\
1
\end{array} \right)
= \left( \begin{array}{c}
3 \\
-1
\end{array} \right)
\][/tex]
The rotated coordinates for this point are [tex]\((3, -1)\)[/tex].
- For point [tex]\((5, 2)\)[/tex]:
[tex]\[
\left( \begin{array}{cc}
-1 & 0 \\
0 & -1
\end{array} \right)
\left( \begin{array}{c}
5 \\
2
\end{array} \right)
= \left( \begin{array}{c}
-5 \\
-2
\end{array} \right)
\][/tex]
The rotated coordinates for this point are [tex]\((-5, -2)\)[/tex].
4. Combine the Results:
Combining all the rotated points, we get the new coordinates of the triangle after the rotation:
[tex]\[
\left(\begin{array}{ccc}
0 & 3 & -5 \\
0 & -1 & -2
\end{array}\right)
\][/tex]
So, the coordinates of the triangle after a [tex]\(180^{\circ}\)[/tex] counter-clockwise rotation about the origin are:
[tex]\[
\left(\begin{array}{ccc}
0 & 3 & -5 \\
0 & -1 & -2
\end{array}\right)
\][/tex]
These are the new positions of the vertices of the triangle after rotating it.
