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A polygon [tex]\( ABCD \)[/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]\(\frac{3}{5}\)[/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]\( A^{\prime}(5.8, -3), B^{\prime}(1.6, -1.5), C^{\prime}(-1.6, 3), D^{\prime}(2.5, 3) \)[/tex]

B. [tex]\( A^{\prime}(-12, 18), B^{\prime}(-6, 6), C^{\prime}(12, -6), D^{\prime}(12, 12) \)[/tex]

C. [tex]\( A^{\prime}(2.4, -3.6), B^{\prime}(1.2, -1.2), C^{\prime}(-2.4, 1.26), D^{\prime}(-2.4, -2.4) \)[/tex]

D. [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]



Answer :

GinnyAnswer
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]

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