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[tex]$\triangle ABC$[/tex] with vertices [tex]$A(-3,0)$[/tex], [tex]$B(-2,3)$[/tex], [tex]$C(-1,1)$[/tex] is rotated [tex]$180^{\circ}$[/tex] clockwise about the origin. It is then reflected across the line [tex]$y=-x$[/tex]. What are the coordinates of the vertices of the image?

A. [tex]$A^{\prime}(0,3)$[/tex], [tex]$B^{\prime}(2,3)$[/tex], [tex]$C(1,1)$[/tex]
B. [tex]$A^{\prime}(0,-3)$[/tex], [tex]$B(3,-2)$[/tex], [tex]$C^{\prime}(1,-1)$[/tex]
C. [tex]$A^{\prime}(-3,0)$[/tex], [tex]$B^{\prime}(-3,2)$[/tex], [tex]$C^{\prime}(-1,1)$[/tex]
D. [tex]$A^{\prime}(0,-3)$[/tex], [tex]$B^{\prime}(-2,-3)$[/tex], [tex]$C^{\prime}(-1,-1)$[/tex]

Sagot :

GinnyAnswer
To solve this problem, we need to apply two transformations to the given triangle vertices [tex]\( A(-3,0), B(-2,3), C(-1,1) \)[/tex]:

1. Rotate [tex]\( \triangle ABC \)[/tex] by [tex]\( 180^\circ \)[/tex] clockwise about the origin.
2. Reflect the resulting image across the line [tex]\( y = -x \)[/tex].

Step 1: Rotate by [tex]\( 180^\circ \)[/tex] clockwise
When a point [tex]\((x, y)\)[/tex] is rotated by [tex]\( 180^\circ \)[/tex] clockwise about the origin, the new coordinates [tex]\((x', y')\)[/tex] will be [tex]\((-x, -y)\)[/tex].

Thus, we have:
- [tex]\( A(-3, 0) \)[/tex] rotated by [tex]\( 180^\circ \)[/tex] becomes [tex]\( A' (3, 0) \)[/tex].
- [tex]\( B(-2, 3) \)[/tex] rotated by [tex]\( 180^\circ \)[/tex] becomes [tex]\( B' (2, -3) \)[/tex].
- [tex]\( C(-1, 1) \)[/tex] rotated by [tex]\( 180^\circ \)[/tex] becomes [tex]\( C' (1, -1) \)[/tex].

Step 2: Reflect across the line [tex]\( y = -x \)[/tex]
To reflect a point [tex]\((x, y)\)[/tex] across the line [tex]\( y = -x \)[/tex], the new coordinates [tex]\((x', y')\)[/tex] will be [tex]\((-y, -x)\)[/tex].

Applying this reflection:
- [tex]\( A' (3, 0) \)[/tex] reflected across [tex]\( y = -x \)[/tex] becomes [tex]\( A'' (0, -3) \)[/tex].
- [tex]\( B' (2, -3) \)[/tex] reflected across [tex]\( y = -x \)[/tex] becomes [tex]\( B'' (3, -2) \)[/tex].
- [tex]\( C' (1, -1) \)[/tex] reflected across [tex]\( y = -x \)[/tex] becomes [tex]\( C'' (1, -1) \)[/tex].

Thus, the coordinates of the vertices after both transformations are:
- [tex]\( A'' (0, -3) \)[/tex]
- [tex]\( B'' (3, -2) \)[/tex]
- [tex]\( C'' (1, -1) \)[/tex]

Therefore, the correct answer is:
R. [tex]\(A^{\prime}(0,-3), B(3,-2), C^{\prime}(1,-1)\)[/tex]
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