To determine the vertices of the image [tex]\( U'V'W' \)[/tex] after rotating the preimage [tex]\( UVW \)[/tex] 90 degrees counterclockwise, follow these steps:
1. Understand the rotation rule:
For a point [tex]\((x, y)\)[/tex], a 90 degrees counterclockwise rotation transforms it to [tex]\((-y, x)\)[/tex].
2. Apply the rotation rule to each vertex:
- For [tex]\( U(-2, 0) \)[/tex]:
[tex]\[
(x, y) = (-2, 0) \implies (-y, x) = (0, -2)
\][/tex]
So, [tex]\( U' \)[/tex] is [tex]\( (0, -2) \)[/tex].
- For [tex]\( V(-3, 1) \)[/tex]:
[tex]\[
(x, y) = (-3, 1) \implies (-y, x) = (-1, -3)
\][/tex]
So, [tex]\( V' \)[/tex] is [tex]\( (-1, -3) \)[/tex].
- For [tex]\( W(-3, 3) \)[/tex]:
[tex]\[
(x, y) = (-3, 3) \implies (-y, x) = (-3, -3)
\][/tex]
So, [tex]\( W' \)[/tex] is [tex]\( (-3, -3) \)[/tex].
3. Compile the new vertices:
The vertices of the image [tex]\( U'V'W' \)[/tex] after the 90 degrees counterclockwise rotation are:
[tex]\[
U'(0, -2), V'(-1, -3), W'(-3, -3)
\][/tex]
Hence, the correct answer is:
[tex]\[
U^{\prime}(0,-2), V^{\prime}(-1,-3), W^{\prime}(-3,-3)
\][/tex]
