To determine the scale factor by which the line segment [tex]\(\overline{AB}\)[/tex] was dilated from the origin to create [tex]\(\overline{A^{\prime}B^{\prime}}\)[/tex], we can use the information provided about the coordinates of the points after the dilation and analyze the distances involved.
Given the points [tex]\(A^{\prime}(0,8)\)[/tex] and [tex]\(B^{\prime}(8,12)\)[/tex]:
1. Identify coordinates after dilation:
- [tex]\(A^{\prime} = (0, 8)\)[/tex]
- [tex]\(B^{\prime} = (8, 12)\)[/tex]
2. Calculate the distances of [tex]\(A'\)[/tex] and [tex]\(B'\)[/tex] from the origin:
- The distance from the origin to [tex]\(A^{\prime}(0,8)\)[/tex] is given by:
[tex]\[
\text{Distance}_{A^{\prime}} = \sqrt{0^2 + 8^2} = \sqrt{64} = 8
\][/tex]
- The distance from the origin to [tex]\(B^{\prime}(8,12)\)[/tex] is given by:
[tex]\[
\text{Distance}_{B^{\prime}} = \sqrt{8^2 + 12^2} = \sqrt{64 + 144} = \sqrt{208} = 4\sqrt{13}
\][/tex]
3. Relate these distances to the scale factor:
- Dilation involves a uniform scale factor [tex]\(k\)[/tex] such that the distances of points from the origin are scaled by [tex]\(k\)[/tex].
- Let's consider that point [tex]\(A\)[/tex] was originally at [tex]\((0, y)\)[/tex]. After dilation, the [tex]\(y\)[/tex]-coordinate of [tex]\(A\)[/tex] becomes 8, so:
[tex]\[
k \cdot y = 8
\][/tex]
- To solve for [tex]\(k\)[/tex], we know the origin distance must equate to the outcome:
For point [tex]\(A\)[/tex]:
[tex]\[
k = \frac{8}{y}
\][/tex]
4. Determining the original coordinate:
- Assume [tex]\(y = 1\)[/tex]. Therefore, [tex]\(k \cdot 1 = 8\)[/tex], we can directly find [tex]\(k = 8\)[/tex].
This analysis shows that the scale factor [tex]\(k\)[/tex] for the dilation is:
[tex]\[
k = 8
\][/tex]
Therefore, the scale factor is:
[tex]\[
~\boxed{8}
\][/tex]
