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Question 10 (Multiple Choice Worth 2 points)

Triangle [tex]$UVW$[/tex] is drawn with vertices at [tex]$U(-1,1)$[/tex], [tex]$V(0,-4)$[/tex], and [tex]$W(-4,-1)$[/tex]. Determine the coordinates of the vertices for the image, triangle [tex]$U'V'W'$[/tex], if the preimage is rotated [tex]$180^{\circ}$[/tex] counterclockwise.

A. [tex]$U'(1,-1)$[/tex], [tex]$V'(0,4)$[/tex], [tex]$W'(4,1)$[/tex]
B. [tex]$U'(-1,-1)$[/tex], [tex]$V'(4,0)$[/tex], [tex]$W'(1,4)$[/tex]
C. [tex]$U'(-1,1)$[/tex], [tex]$V'(4,0)$[/tex], [tex]$W'(1,-4)$[/tex]
D. [tex]$U'(-1,1)$[/tex], [tex]$V'(0,-4)$[/tex], [tex]$W(-4,-1)$[/tex]



Answer :

GinnyAnswer
To determine the coordinates of the vertices for the image of triangle [tex]\( U V W \)[/tex] after a [tex]\( 180^\circ \)[/tex] counterclockwise rotation, we need to follow the properties of coordinates transformation for such a rotation.

When a point [tex]\((x, y)\)[/tex] is rotated [tex]\( 180^\circ \)[/tex] counterclockwise around the origin, the new coordinates become [tex]\((-x, -y)\)[/tex].

Given the original vertices of triangle [tex]\( U V W \)[/tex]:
[tex]\[ U(-1, 1) \][/tex]
[tex]\[ V(0, -4) \][/tex]
[tex]\[ W(-4, -1) \][/tex]

We can transform each vertex as follows:

1. For vertex [tex]\( U(-1, 1) \)[/tex]:
[tex]\[ U' = (-(-1), -(1)) = (1, -1) \][/tex]

2. For vertex [tex]\( V(0, -4) \)[/tex]:
[tex]\[ V' = (-(0), -(-4)) = (0, 4) \][/tex]

3. For vertex [tex]\( W(-4, -1) \)[/tex]:
[tex]\[ W' = (-(-4), -(-1)) = (4, 1) \][/tex]

Thus, the new coordinates of the vertices for the image triangle [tex]\( U' V' W' \)[/tex] after a [tex]\( 180^\circ \)[/tex] counterclockwise rotation are:
[tex]\[ U' (1, -1) \][/tex]
[tex]\[ V' (0, 4) \][/tex]
[tex]\[ W' (4, 1) \][/tex]

The correct answer is:
[tex]\[ U' (1, -1), V' (0, 4), W' (4, 1) \][/tex]

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