To determine the coordinates of polygon [tex]\(A^{\prime} B^{\prime} C^{\prime} D^{\prime}\)[/tex] after dilation, we'll dilate each vertex of polygon [tex]\(ABCD\)[/tex] by a scale factor of [tex]\( \frac{3}{5} \)[/tex] with the center of dilation at the origin [tex]\((0,0)\)[/tex].
The dilation formula for any point [tex]\((x, y)\)[/tex] with respect to the origin using a scale factor [tex]\(k\)[/tex] is:
[tex]\[ (x', y') = (kx, ky) \][/tex]
Given:
- Original vertices:
- [tex]\(A(-4, 6)\)[/tex]
- [tex]\(B(-2, 2)\)[/tex]
- [tex]\(C(4, -2)\)[/tex]
- [tex]\(D(4, 4)\)[/tex]
- Scale factor: [tex]\( \frac{3}{5} \)[/tex]
Let's apply the dilation to each vertex step-by-step:
1. For vertex [tex]\(A(-4, 6)\)[/tex]:
[tex]\[
\begin{align*}
x' &= \frac{3}{5} \cdot (-4) = -\frac{12}{5} = -2.4 \\
y' &= \frac{3}{5} \cdot 6 = \frac{18}{5} = 3.6 \\
A' &= (-2.4, 3.6)
\end{align*}
\][/tex]
2. For vertex [tex]\(B(-2, 2)\)[/tex]:
[tex]\[
\begin{align*}
x' &= \frac{3}{5} \cdot (-2) = -\frac{6}{5} = -1.2 \\
y' &= \frac{3}{5} \cdot 2 = \frac{6}{5} = 1.2 \\
B' &= (-1.2, 1.2)
\end{align*}
\][/tex]
3. For vertex [tex]\(C(4, -2)\)[/tex]:
[tex]\[
\begin{align*}
x' &= \frac{3}{5} \cdot 4 = \frac{12}{5} = 2.4 \\
y' &= \frac{3}{5} \cdot (-2) = -\frac{6}{5} = -1.2 \\
C' &= (2.4, -1.2)
\end{align*}
\][/tex]
4. For vertex [tex]\(D(4, 4)\)[/tex]:
[tex]\[
\begin{align*}
x' &= \frac{3}{5} \cdot 4 = \frac{12}{5} = 2.4 \\
y' &= \frac{3}{5} \cdot 4 = \frac{12}{5} = 2.4 \\
D' &= (2.4, 2.4)
\end{align*}
\][/tex]
After calculating these, you will find the new coordinates of the vertices for polygon [tex]\( A^{\prime} B^{\prime} C^{\prime} D^{\prime} \)[/tex]:
[tex]\[
A^{\prime} = (-2.4, 3.6), \quad B^{\prime} = (-1.2, 1.2), \quad C^{\prime} = (2.4, -1.2), \quad D^{\prime} = (2.4, 2.4)
\][/tex]
Hence, the correct set of vertices for polygon [tex]\( A^{\prime} B^{\prime} C^{\prime} D^{\prime} \)[/tex] is:
[tex]\[
\boxed{A^{\prime}(-2.4, 3.6), B^{\prime}(-1.2, 1.2), C^{\prime}(2.4, -1.2), D^{\prime}(2.4, 2.4)}
\][/tex]
