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Triangle [tex]$XYZ$[/tex] has vertices [tex]$X(1,-1)$[/tex], [tex]$Y(3,4)$[/tex], and [tex]$Z(5,-1)$[/tex]. Nikko rotated the triangle [tex]$90^{\circ}$[/tex] counterclockwise about the origin. What is the correct set of image points for triangle [tex]$X^{\prime}Y^{\prime}Z^{\prime}$[/tex]?

A. [tex]$X^{\prime}(1,1)$[/tex], [tex]$Y^{\prime}(-4,3)$[/tex], [tex]$Z^{\prime}(1,5)$[/tex]
B. [tex]$X^{\prime}(1,1)$[/tex], [tex]$Y^{\prime}(-3,-4)$[/tex], [tex]$Z^{\prime}(-1,-5)$[/tex]
C. [tex]$X^{\prime}(-1,1)$[/tex], [tex]$Y^{\prime}(-3,-4)$[/tex], [tex]$Z^{\prime}(-5,1)$[/tex]
D. [tex]$X^{\prime}(-1,-1)$[/tex], [tex]$Y^{\prime}(4,-3)$[/tex], [tex]$Z^{\prime}(-1,-5)$[/tex]

Sagot :

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].