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Question 7 (Multiple Choice Worth 5 points)

Polygon [tex]$A B C D$[/tex] with vertices at [tex]$A(-4,6)$[/tex], [tex]$B(-2,2)$[/tex], [tex]$C(4,-2)$[/tex], and [tex]$D(4,4)$[/tex] is dilated using a scale factor of [tex]$\frac{3}{8}$[/tex] to create polygon [tex]$A^{\prime} B^{\prime} C^{\prime} D^{\prime}$[/tex]. If the dilation is centered at the origin, determine the vertices of polygon [tex]$A^{\prime} B^{\prime} C^{\prime} D^{\prime}$[/tex].

A. [tex]$A^{\prime}(-1.5,2.25)$[/tex], [tex]$B^{\prime}(-0.75,0.75)$[/tex], [tex]$C^{\prime}(1.5,-0.75)$[/tex], [tex]$D^{\prime}(1.5,1.5)$[/tex]

B. [tex]$A^{\prime}(-12,18)$[/tex], [tex]$B^{\prime}(-6,6)$[/tex], [tex]$C^{\prime}(12,-6)$[/tex], [tex]$D^{\prime}(12,12)$[/tex]

C. [tex]$A^{\prime}(0.75,-1.25)$[/tex], [tex]$B^{\prime}(1.6,-1.6)$[/tex], [tex]$C^{\prime}(-1.25,1.6)$[/tex], [tex]$D^{\prime}(-0.075,-0.75)$[/tex]

D. [tex]$A^{\prime}(1.27,-3)$[/tex], [tex]$B^{\prime}(3.2,-3.2)$[/tex], [tex]$C^{\prime}(-0.75,3)$[/tex], [tex]$D^{\prime}(3,3)$[/tex]



Answer :

GinnyAnswer
To determine the vertices of the dilated polygon [tex]\( A'B'C'D' \)[/tex] given a scale factor of [tex]\(\frac{3}{8}\)[/tex] and the origin as the center of dilation, follow these steps:

1. Identify Original Vertices:
The original vertices of polygon [tex]\(ABCD\)[/tex] are:
- [tex]\( A(-4, 6) \)[/tex]
- [tex]\( B(-2, 2) \)[/tex]
- [tex]\( C(4, -2) \)[/tex]
- [tex]\( D(4, 4) \)[/tex]

2. Apply Dilations:
To find the coordinates of the vertices [tex]\(A'(-1.5, 2.25)\)[/tex], [tex]\(B'(-0.75, 0.75)\)[/tex], [tex]\(C'(1.5, -0.75)\)[/tex], and [tex]\(D'(1.5, 1.5)\)[/tex] after dilation, we multiply each coordinate of the original vertices by the scale factor [tex]\(\frac{3}{8}\)[/tex].

- For vertex [tex]\(A\)[/tex]:
[tex]\[ A' = \left(\frac{3}{8} \times (-4), \frac{3}{8} \times 6\right) = (-1.5, 2.25) \][/tex]

- For vertex [tex]\(B\)[/tex]:
[tex]\[ B' = \left(\frac{3}{8} \times (-2), \frac{3}{8} \times 2\right) = (-0.75, 0.75) \][/tex]

- For vertex [tex]\(C\)[/tex]:
[tex]\[ C' = \left(\frac{3}{8} \times 4, \frac{3}{8} \times (-2)\right) = (1.5, -0.75) \][/tex]

- For vertex [tex]\(D\)[/tex]:
[tex]\[ D' = \left(\frac{3}{8} \times 4, \frac{3}{8} \times 4\right) = (1.5, 1.5) \][/tex]

3. Compare with Multiple Choices:
The vertices after dilation are:
- [tex]\( A'(-1.5, 2.25) \)[/tex]
- [tex]\( B'(-0.75, 0.75) \)[/tex]
- [tex]\( C'(1.5, -0.75) \)[/tex]
- [tex]\( D'(1.5, 1.5) \)[/tex]

By comparing the provided options, the correct choice is:

[tex]\[ \boxed{A^{\prime}(-1.5,2.25), B^{\prime}(-0.75,0.75), C^{\prime}(1.5,-0.75), D^{\prime}(1.5,1.5)} \][/tex]

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