To determine the vertices of the image [tex]\( U^{\prime} V^{\prime} W^{\prime} \)[/tex] after rotating the triangle [tex]\( UVW \)[/tex] [tex]\( 90^\circ \)[/tex] clockwise, follow these steps:
1. Understand the rotation rule: A [tex]\( 90^\circ \)[/tex] clockwise rotation transforms a point [tex]\((x, y)\)[/tex] to [tex]\((y, -x)\)[/tex].
2. Apply the rotation to each vertex:
- Vertex [tex]\( U(-2, 0) \)[/tex]:
- Original coordinates: [tex]\((-2, 0)\)[/tex]
- Applying the rule: [tex]\((0, -(-2)) = (0, 2)\)[/tex]
- Therefore, [tex]\( U' = (0, 2) \)[/tex].
- Vertex [tex]\( V(-3, 1) \)[/tex]:
- Original coordinates: [tex]\((-3, 1)\)[/tex]
- Applying the rule: [tex]\((1, -(-3)) = (1, 3)\)[/tex]
- Therefore, [tex]\( V' = (1, 3) \)[/tex].
- Vertex [tex]\( W(-3, 3) \)[/tex]:
- Original coordinates: [tex]\((-3, 3)\)[/tex]
- Applying the rule: [tex]\((3, -(-3)) = (3, 3)\)[/tex]
- Therefore, [tex]\( W' = (3, 3) \)[/tex].
3. Summarize the results:
- The vertices of the image [tex]\( U^{\prime} V^{\prime} W^{\prime} \)[/tex] after a [tex]\( 90^\circ \)[/tex] clockwise rotation are:
- [tex]\( U' = (0, 2) \)[/tex]
- [tex]\( V' = (1, 3) \)[/tex]
- [tex]\( W' = (3, 3) \)[/tex]
Given the options, the correct set of vertices after the rotation is:
[tex]\[ \boxed{U ^{\prime}(0, 2), V ^{\prime}(1, 3), W ^{\prime}(3, 3)} \][/tex]
