To find the length of segment [tex]\(A'B'\)[/tex] after scaling down the quadrilateral with a scale factor of [tex]\(\frac{1}{4}\)[/tex], we need to follow these steps:
1. Identify the coordinates of points A and B:
- [tex]\(A(0,4)\)[/tex]
- [tex]\(B(8,4)\)[/tex]
2. Calculate the original length of segment [tex]\(AB\)[/tex]:
- Using the distance formula for two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex], the distance [tex]\(d\)[/tex] is given by:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
- Plugging in the coordinates of [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
[tex]\[
AB = \sqrt{(8 - 0)^2 + (4 - 4)^2}
\][/tex]
[tex]\[
AB = \sqrt{8^2 + 0^2}
\][/tex]
[tex]\[
AB = \sqrt{64}
\][/tex]
[tex]\[
AB = 8
\][/tex]
3. Apply the scale factor [tex]\(\frac{1}{4}\)[/tex]:
- The length of [tex]\(A'B'\)[/tex] after applying the scale factor is:
[tex]\[
A'B' = AB \times \frac{1}{4}
\][/tex]
- Substituting the length of [tex]\(AB\)[/tex]:
[tex]\[
A'B' = 8 \times \frac{1}{4}
\][/tex]
[tex]\[
A'B' = 2
\][/tex]
Thus, the length of segment [tex]\(A'B'\)[/tex] after scaling is [tex]\(2\)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{2} \][/tex]
