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.
