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Question 1 (Multiple Choice, Worth 2 points)

Polygon [tex]\( ABCD \)[/tex] with vertices at [tex]\( A(-4, 6), B(-2, 2), C(4, -2), D(4, 4) \)[/tex] is dilated using a scale factor of [tex]\(\frac{1}{4}\)[/tex] to create polygon [tex]\( A'B'C'D' \)[/tex]. Determine the vertices of polygon [tex]\( A'B'C'D' \)[/tex].

A. [tex]\( A'(-0.8, 1.2), B'(-0.4, 0.4), C'(0.8, -0.4), D'(0.8, 0.8) \)[/tex]

B. [tex]\( A'(-1, 1.5), B'(-0.5, 0.5), C'(1, -0.5), D'(1, 1) \)[/tex]

C. [tex]\( A'(-2, 3), B'(-1, 1), C'(2, -1), D'(2, 2) \)[/tex]

D. [tex]\( A'(-3, 4.5), B'(-1.5, 1.5), C'(3, -1.5), D'(3, 3) \)[/tex]

Question 2 (Multiple Choice, Worth 2 points)

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Answer :

GinnyAnswer
To solve the problem of dilating polygon [tex]$A B C D$[/tex] with given vertices by a scale factor of [tex]$\frac{1}{4}$[/tex], let's go through it step-by-step:

1. Identify the original coordinates:
- [tex]$A(-4, 6)$[/tex]
- [tex]$B(-2, 2)$[/tex]
- [tex]$C(4, -2)$[/tex]
- [tex]$D(4, 4)$[/tex]

2. Apply the scale factor:
- The coordinates of the new vertices [tex]$A^{\prime}$[/tex], [tex]$B^{\prime}$[/tex], [tex]$C^{\prime}$[/tex], and [tex]$D^{\prime}$[/tex] are found by multiplying each coordinate by [tex]$\frac{1}{4}$[/tex].

3. Calculate the new coordinates:
- For [tex]$A$[/tex]:
[tex]\[ A^{\prime} = \left(\frac{-4}{4}, \frac{6}{4}\right) = (-1.0, 1.5) \][/tex]
- For [tex]$B$[/tex]:
[tex]\[ B^{\prime} = \left(\frac{-2}{4}, \frac{2}{4}\right) = (-0.5, 0.5) \][/tex]
- For [tex]$C$[/tex]:
[tex]\[ C^{\prime} = \left(\frac{4}{4}, \frac{-2}{4}\right) = (1.0, -0.5) \][/tex]
- For [tex]$D$[/tex]:
[tex]\[ D^{\prime} = \left(\frac{4}{4}, \frac{4}{4}\right) = (1.0, 1.0) \][/tex]

4. Summarize the new vertices:
[tex]\[ A^{\prime} = (-1.0, 1.5) \][/tex]
[tex]\[ B^{\prime} = (-0.5, 0.5) \][/tex]
[tex]\[ C^{\prime} = (1.0, -0.5) \][/tex]
[tex]\[ D^{\prime} = (1.0, 1.0) \][/tex]

Therefore, the vertices of the dilated polygon [tex]$A^{\prime} B^{\prime} C^{\prime} D^{\prime}$[/tex] are:
[tex]\(( -1, 1.5)\)[/tex], [tex]\(( -0.5, 0.5)\)[/tex], [tex]\((1, -0.5)\)[/tex], and [tex]\((1, 1)\)[/tex].

The correct answer to the multiple-choice question is:
[tex]\[ \boxed{A^{\prime}(-1,1.5), B^{\prime}(-0.5,0.5), C^{\prime}(1,-0.5), D^{\prime}(1,1)} \][/tex]