To determine the correct set of image points for triangle [tex]\( X' Y' Z' \)[/tex] after rotating triangle [tex]\( XYZ \)[/tex] 90 degrees counterclockwise about the origin, let's first understand the geometric transformation involved.
A 90-degree counterclockwise rotation about the origin can be carried out using the following rule:
- If a point [tex]\( (x, y) \)[/tex] is rotated 90 degrees counterclockwise about the origin, its new coordinates will be [tex]\( (-y, x) \)[/tex].
Let’s apply this transformation to each vertex of the triangle [tex]\( XYZ \)[/tex]:
1. Vertex [tex]\( X(1, -1) \)[/tex]:
- Initial coordinates: [tex]\( (1, -1) \)[/tex]
- After rotation: [tex]\( (-(-1), 1) = (1, 1) \)[/tex]
- So, [tex]\( X' = (1, 1) \)[/tex]
2. Vertex [tex]\( Y(3, 4) \)[/tex]:
- Initial coordinates: [tex]\( (3, 4) \)[/tex]
- After rotation: [tex]\( (-4, 3) \)[/tex]
- So, [tex]\( Y' = (-4, 3) \)[/tex]
3. Vertex [tex]\( Z(5, -1) \)[/tex]:
- Initial coordinates: [tex]\( (5, -1) \)[/tex]
- After rotation: [tex]\( (-(-1), 5) = (1, 5) \)[/tex]
- So, [tex]\( Z' = (1, 5) \)[/tex]
Therefore, the new coordinates of the triangle vertices after the 90-degree counterclockwise rotation are:
[tex]\[ X' = (1, 1), \, Y' = (-4, 3), \, Z' = (1, 5) \][/tex]
This corresponds to the first option provided:
[tex]\[ X(1,1), Y(-4,3), Z(1,5) \][/tex]
So, the correct set of image points for triangle [tex]\( X' Y' Z' \)[/tex] is:
[tex]\[ (1,1), (-4,3), (1,5) \][/tex]
