To determine the coordinates of the vertices for the image of triangle UVW after a [tex]\(90^\circ\)[/tex] counterclockwise rotation, let's look at each vertex one by one:
1. Vertex [tex]\(U(-1, 1)\)[/tex]:
- Rotating [tex]\(U\)[/tex] by [tex]\(90^\circ\)[/tex] counterclockwise, we swap the coordinates and then change the sign of the new x-coordinate. Thus, the new coordinates are:
[tex]\[
U' = (-1, -1)
\][/tex]
2. Vertex [tex]\(V(0, -4)\)[/tex]:
- Similarly, rotating [tex]\(V\)[/tex] by [tex]\(90^\circ\)[/tex] counterclockwise, we swap the coordinates and change the sign of the new x-coordinate. So the coordinates become:
[tex]\[
V' = (4, 0)
\][/tex]
3. Vertex [tex]\(W(-4, -1)\)[/tex]:
- For vertex [tex]\(W\)[/tex], after a [tex]\(90^\circ\)[/tex] counterclockwise rotation, we swap the coordinates and change the sign of the new x-coordinate. Therefore, the coordinates turn into:
[tex]\[
W' = (1, -4)
\][/tex]
So, the coordinates of the vertices for the image, triangle [tex]\(U'V'W'\)[/tex], after the [tex]\(90^\circ\)[/tex] counterclockwise rotation are:
[tex]\[
U' = (-1, -1), \quad V' = (4, 0), \quad W' = (1, -4)
\][/tex]
Hence, the correct option is:
[tex]\[
\boxed{U'(-1,-1), \, V'(4,0), \, W'(1,-4)}
\][/tex]
