First, let's understand what happens to a point when it is rotated [tex]\(90^{\circ}\)[/tex] counterclockwise about the origin. The original coordinates [tex]\((x, y)\)[/tex] of a point are transformed to [tex]\((-y, x)\)[/tex].
### Given Points:
1. [tex]\( X(1, -1) \)[/tex]
2. [tex]\( Y(3, 4) \)[/tex]
3. [tex]\( Z(5, -1) \)[/tex]
### Step-by-Step Rotation:
1. Rotate [tex]\( X(1, -1) \)[/tex]:
- Original coordinates: [tex]\( (1, -1) \)[/tex]
- New coordinates after [tex]\(90^{\circ}\)[/tex] counterclockwise rotation: [tex]\((-(-1), 1) = (1, 1)\)[/tex]
2. Rotate [tex]\( Y(3, 4) \)[/tex]:
- Original coordinates: [tex]\( (3, 4) \)[/tex]
- New coordinates after [tex]\(90^{\circ}\)[/tex] counterclockwise rotation: [tex]\((-4, 3)\)[/tex]
3. Rotate [tex]\( Z(5, -1) \)[/tex]:
- Original coordinates: [tex]\( (5, -1) \)[/tex]
- New coordinates after [tex]\(90^{\circ}\)[/tex] counterclockwise rotation: [tex]\((-(-1), 5) = (1, 5)\)[/tex]
### Result:
The image points [tex]\(X^{\prime}, Y^{\prime}, Z^{\prime}\)[/tex] after the [tex]\(90^{\circ}\)[/tex] counterclockwise rotation are:
- [tex]\(X^{\prime}(1, 1)\)[/tex]
- [tex]\(Y^{\prime}(-4, 3)\)[/tex]
- [tex]\(Z^{\prime}(1, 5)\)[/tex]
Therefore, the correct set of image points for the triangle [tex]\(X^{\prime}Y^{\prime}Z^{\prime}\)[/tex] is [tex]\(\boxed{X^{\prime}(1, 1), Y^{\prime}(-4, 3), Z^{\prime}(1, 5)}\)[/tex].
