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Triangle UVW is drawn with vertices at [tex]\( U(-1,1), V(0,-4), W(-4,-1) \)[/tex]. Determine the coordinates of the vertices for the image, triangle [tex]\( UVW \)[/tex], if the preimage is rotated [tex]\( 90^{\circ} \)[/tex] counterclockwise.

A. [tex]\( U(1,-1), V(0,4), W(4,1) \)[/tex]

B. [tex]\( U^{\prime}(-1,-1), V(4,0), W(1,-4) \)[/tex]

C. [tex]\( U(-1,1), V(4,0), W(1,4) \)[/tex]

D. [tex]\( U^{\prime}(-1,1), V(0,-4), W(-4,-1) \)[/tex]



Answer :

GinnyAnswer
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]

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