To determine which statement accurately describes the pre-image segment [tex]\(\overline{Y Z}\)[/tex] given the endpoints of the dilated segment [tex]\(\overline{Y^{\prime} Z^{\prime}}\)[/tex] and a scale factor of 3 from the origin, follow these steps:
1. Identify the given points of [tex]\(\overline{Y^{\prime} Z^{\prime}}\)[/tex]:
- [tex]\(Y^{\prime} = (0, 3)\)[/tex]
- [tex]\(Z^{\prime} = (-6, 3)\)[/tex]
2. Recall the scale factor:
- Scale factor [tex]\(= 3\)[/tex]
3. Understand that dilation by a scale factor [tex]\(k\)[/tex] from the origin transforms a point [tex]\((x, y)\)[/tex] to [tex]\((kx, ky)\)[/tex].
To find the pre-image coordinates [tex]\((x, y)\)[/tex], given the image coordinates [tex]\((x', y')\)[/tex], we divide by the scale factor [tex]\(k\)[/tex].
4. Calculate the pre-image coordinates:
- For [tex]\(Y^{\prime} = (0, 3)\)[/tex], the pre-image [tex]\((Y)\)[/tex] coordinates are:
[tex]\[
x = \frac{0}{3} = 0
\][/tex]
[tex]\[
y = \frac{3}{3} = 1
\][/tex]
So, [tex]\(Y = (0, 1)\)[/tex].
- For [tex]\(Z^{\prime} = (-6, 3)\)[/tex], the pre-image [tex]\((Z)\)[/tex] coordinates are:
[tex]\[
x = \frac{-6}{3} = -2
\][/tex]
[tex]\[
y = \frac{3}{3} = 1
\][/tex]
So, [tex]\(Z = (-2, 1)\)[/tex].
5. Summarize the pre-image coordinates:
- [tex]\(Y = (0, 1)\)[/tex]
- [tex]\(Z = (-2, 1)\)[/tex]
Therefore, the correct statement describing the pre-image segment [tex]\(\overline{Y Z}\)[/tex] is:
[tex]\[
\overline{Y Z} \text{ is located at } Y(0,1) \text{ and } Z(-2,1) \text{ and is one-third the size of } \overline{Y^{\prime} Z^{\prime}}.
\][/tex]
