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Polygon [tex]A B C D[/tex] with vertices at [tex]A(-4,6), B(-2,2), 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), B^{\prime}(-0.75,0.75), C^{\prime}(1.5,-0.75), D^{\prime}(1.5,1.5)[/tex]
B. [tex]A^{\prime}(-12,18), B^{\prime}(-6,6), C^{\prime}(12,-6), D^{\prime}(12,12)[/tex]
C. [tex]A^{\prime}(0.75,-1.25), B^{\prime}(1.6,-1.6), C^{\prime}(-1.25,1.6), D^{\prime}(-0.075,-0.75)[/tex]
D. [tex]A^{\prime}(1.27,-3), B^{\prime}(3,2,-3,2), C^{\prime}(-0.75,3), D^{\prime}(3,3)[/tex]



Answer :

GinnyAnswer
To determine the vertices of polygon [tex]\(A' B' C' D'\)[/tex] after the dilation, let's follow the steps carefully:

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

2. Determine the Scale Factor and Dilation Center:
The scale factor for the dilation is [tex]\(\frac{3}{8}\)[/tex] and the dilation is centered at the origin [tex]\((0, 0)\)[/tex].

3. Apply the Scale Factor to Each Vertex:
We will apply the scale factor [tex]\(\frac{3}{8}\)[/tex] to each coordinate of the vertices.

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

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

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

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

4. List the Vertices after Dilation:
The vertices of the dilated polygon [tex]\(A' B' C' D'\)[/tex] are:
[tex]\[ A'(-1.5, 2.25), \quad B'(-0.75, 0.75), \quad C'(1.5, -0.75), \quad D'(1.5, 1.5) \][/tex]

Thus, the vertices of polygon [tex]\(A' B' C' D'\)[/tex] after dilation are:
[tex]\[ A'(-1.5, 2.25), \quad B'(-0.75, 0.75), \quad C'(1.5, -0.75), \quad D'(1.5, 1.5) \][/tex]

This matches the first choice provided in the question, confirming our result.

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