To solve the problem of finding the vertices of the polygon [tex]\(A'B'C'D'\)[/tex] after dilation, we need to perform the following steps:
1. Identify the original coordinates of the vertices:
- [tex]\(A(-4,6)\)[/tex]
- [tex]\(B(-2,2)\)[/tex]
- [tex]\(C(4,-2)\)[/tex]
- [tex]\(D(4,4)\)[/tex]
2. Use the given scale factor for dilation: [tex]\(\frac{3}{5}\)[/tex].
3. Dilation calculation:
- To dilate a point [tex]\((x, y)\)[/tex] by a scale factor [tex]\(k\)[/tex] centered at the origin, the new coordinates [tex]\((x', y')\)[/tex] are given by:
[tex]\[
x' = k \cdot x
\][/tex]
[tex]\[
y' = k \cdot y
\][/tex]
4. Calculate the new coordinates after dilation:
- For the vertex [tex]\(A (-4, 6)\)[/tex]:
[tex]\[
A' = \left( \frac{3}{5} \cdot (-4), \frac{3}{5} \cdot 6 \right) = (-2.4, 3.6)
\][/tex]
- For the vertex [tex]\(B (-2, 2)\)[/tex]:
[tex]\[
B' = \left( \frac{3}{5} \cdot (-2), \frac{3}{5} \cdot 2 \right) = (-1.2, 1.2)
\][/tex]
- For the vertex [tex]\(C (4, -2)\)[/tex]:
[tex]\[
C' = \left( \frac{3}{5} \cdot 4, \frac{3}{5} \cdot (-2) \right) = (2.4, -1.2)
\][/tex]
- For the vertex [tex]\(D (4, 4)\)[/tex]:
[tex]\[
D' = \left( \frac{3}{5} \cdot 4, \frac{3}{5} \cdot 4 \right) = (2.4, 2.4)
\][/tex]
5. The vertices of the dilated polygon are:
[tex]\[
A'(-2.4, 3.6), B'(-1.2, 1.2), C'(2.4, -1.2), D'(2.4, 2.4)
\][/tex]
Thus, the correct choice, considering the dilation transformation, is:
[tex]\[
A^{\prime}(-2.4, 3.6), B^{\prime}(-1.2, 1.2), C^{\prime}(2.4, -1.2), D^{\prime}(2.4, 2.4)
\][/tex]
This corresponds to the final option provided in the problem:
[tex]\(\boxed{A'(-2.4, 3.6), B'(-1.2, 1.2), C'(2.4, -1.2), D'(2.4, 2.4)}\)[/tex].
