To solve the problem of comparing the length of side [tex]\(AB\)[/tex] with its scaled version [tex]\(A'B'\)[/tex], we will proceed with the following steps:
### Step 1: Determine the length of [tex]\(AB\)[/tex]
Given the coordinates of points [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
- [tex]\(A(0, 4)\)[/tex]
- [tex]\(B(8, 4)\)[/tex]
We use the distance formula to calculate the length of [tex]\(AB\)[/tex]:
[tex]\[
\text{Length of } AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
where [tex]\(A = (x_1, y_1)\)[/tex] and [tex]\(B = (x_2, y_2)\)[/tex].
Substituting the coordinates:
[tex]\[
\text{Length of } AB = \sqrt{(8 - 0)^2 + (4 - 4)^2}
\][/tex]
[tex]\[
= \sqrt{8^2 + 0^2}
\][/tex]
[tex]\[
= \sqrt{64}
\][/tex]
[tex]\[
= 8.0
\][/tex]
### Step 2: Determine the length of [tex]\(A'B'\)[/tex]
Given the scale factor [tex]\(\frac{1}{4}\)[/tex], the length of the scaled side [tex]\(A'B'\)[/tex] is:
[tex]\[
\text{Length of } A'B' = \text{Length of } AB \times \frac{1}{4}
\][/tex]
[tex]\[
= 8.0 \times \frac{1}{4}
\][/tex]
[tex]\[
= 2.0
\][/tex]
### Step 3: Determine the ratio of the lengths
To compare the lengths of [tex]\(AB\)[/tex] and [tex]\(A'B'\)[/tex], we find the ratio of the length of [tex]\(AB\)[/tex] to the length of [tex]\(A'B'\)[/tex]:
[tex]\[
\text{Ratio} = \frac{\text{Length of } AB}{\text{Length of } A'B'}
\][/tex]
[tex]\[
= \frac{8.0}{2.0}
\][/tex]
[tex]\[
= 4.0
\][/tex]
### Conclusion
The relationship between the length of side [tex]\(AB\)[/tex] and the length of side [tex]\(A'B'\)[/tex] is that the length of [tex]\(A'B'\)[/tex] is exactly one-fourth the length of [tex]\(AB\)[/tex]. The ratio of the length of [tex]\(AB\)[/tex] to the length of [tex]\(A'B'\)[/tex] is [tex]\(4.0\)[/tex].
