To determine the correct pre-image of the segment [tex]\(\overline{E'F'}\)[/tex] after dilation by a scale factor of [tex]\(\frac{1}{2}\)[/tex] from the origin, follow these steps:
1. Identify the given points after dilation:
- [tex]\(E'(1, 0)\)[/tex]
- [tex]\(F'(1, 3)\)[/tex]
2. Understand the dilation transformation:
- A dilation by a scale factor of [tex]\(\frac{1}{2}\)[/tex] means that the pre-image coordinates are scaled down by a factor of 2 to obtain the image coordinates. Therefore, to find the pre-image coordinates, we must multiply the image coordinates by 2.
3. Calculate the pre-image coordinates:
- For [tex]\(E'\)[/tex] with coordinates [tex]\((1, 0)\)[/tex], the pre-image [tex]\(E\)[/tex] is calculated as:
[tex]\[
E = \left( 1 \div \frac{1}{2}, 0 \div \frac{1}{2} \right) = \left( 2 \times 1, 2 \times 0 \right) = (2, 0)
\][/tex]
- For [tex]\(F'\)[/tex] with coordinates [tex]\((1, 3)\)[/tex], the pre-image [tex]\(F\)[/tex] is calculated as:
[tex]\[
F = \left( 1 \div \frac{1}{2}, 3 \div \frac{1}{2} \right) = \left( 2 \times 1, 2 \times 3 \right) = (2, 6)
\][/tex]
4. Compare the options and confirm the correct one:
- The pre-image coordinates [tex]\(E(2, 0)\)[/tex] and [tex]\(F(2, 6)\)[/tex] fit the description given in one of the statements. Additionally, since the pre-image was dilated by a factor of [tex]\(\frac{1}{2}\)[/tex], the pre-image segment [tex]\(\overline{EF}\)[/tex] must be twice the size of the image segment [tex]\(\overline{E'F'}\)[/tex].
5. Select the correct statement:
The correct statement based on the calculated pre-image coordinates and the effect of the scale factor is:
[tex]\[
\text{``$\overline{EF}$ is located at $E(2,0)$ and $F(2,6)$ and is twice the size of $\overline{E'F'}$.''}
\][/tex]
Thus, the pre-image coordinates and their relationship to the image segment make the correct answer:
[tex]\(\boxed{3}\)[/tex]
