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]
