To determine the vertices of the image [tex]\(U'V'W'\)[/tex] after rotating triangle [tex]\(UVW\)[/tex] 180° counterclockwise, we need to apply the rotation transformation to each vertex of the triangle.
A rotation by 180° counterclockwise (which is the same as a 180° clockwise rotation due to symmetry) about the origin can be achieved using the following transformation:
[tex]\[
(x', y') = (-x, -y)
\][/tex]
This means for any vertex [tex]\((x, y)\)[/tex], the new coordinates after the rotation will be [tex]\((-x, -y)\)[/tex].
Let's apply this transformation to each vertex of triangle [tex]\(UVW\)[/tex]:
1. For vertex [tex]\(U(-2, 0)\)[/tex]:
[tex]\[
U' = (-(-2), -0) = (2, 0)
\][/tex]
2. For vertex [tex]\(V(-3, 1)\)[/tex]:
[tex]\[
V' = (-(-3), -1) = (3, -1)
\][/tex]
3. For vertex [tex]\(W(-3, 3)\)[/tex]:
[tex]\[
W' = (-(-3), -3) = (3, -3)
\][/tex]
So, the vertices of the image [tex]\(U'V'W'\)[/tex] after rotating 180° counterclockwise are:
[tex]\[
U'(2, 0), V'(3, -1), W'(3, -3)
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{U'(2, 0), V'(3, -1), W'(3, -3)}
\][/tex]
