To determine the scale factor [tex]\( k \)[/tex] by which line segment [tex]\(\overline{AB}\)[/tex] was dilated from the origin to create [tex]\(\overline{A^{\prime}B^{\prime}}\)[/tex], we follow these steps:
1. Identify the given coordinates:
- [tex]\( A' \)[/tex] has coordinates [tex]\( (0, 8) \)[/tex].
- [tex]\( B' \)[/tex] has coordinates [tex]\( (8, 12) \)[/tex].
2. Apply the properties of dilation:
The coordinates of the image points [tex]\( A' \)[/tex] and [tex]\( B' \)[/tex] are obtained by multiplying the original coordinates of [tex]\( A \)[/tex] and [tex]\( B \)[/tex] by the scale factor [tex]\( k \)[/tex]. Thus, if [tex]\( A(x_1, y_1) \)[/tex] and [tex]\( B(x_2, y_2) \)[/tex] are the coordinates before dilation, then:
- [tex]\( A'(0, 8) = (k \cdot x_1, k \cdot y_1) \)[/tex].
- [tex]\( B'(8, 12) = (k \cdot x_2, k \cdot y_2) \)[/tex].
3. Determine the relationships from given points:
- For point [tex]\( A' \)[/tex]:
[tex]\[
0 = k \cdot x_1 \quad \text{and} \quad 8 = k \cdot y_1
\][/tex]
This indicates [tex]\( x_1 = 0 \)[/tex].
- For point [tex]\( B' \)[/tex]:
[tex]\[
8 = k \cdot x_2 \quad \text{and} \quad 12 = k \cdot y_2
\][/tex]
4. Calculate the scale factor [tex]\( k \)[/tex]:
- From [tex]\( A'(0, 8) \)[/tex]:
[tex]\[
k \cdot y_1 = 8 \rightarrow k = \frac{8}{y_1}
\][/tex]
- From [tex]\( B'(8, 12) \)[/tex]:
[tex]\[
8 = k \cdot x_2 \quad \Rightarrow \quad k = \frac{8}{x_2}
\][/tex]
[tex]\[
12 = k \cdot y_2 \quad \Rightarrow \quad k = \frac{12}{y_2}
\][/tex]
5. Setup common [tex]\( k \)[/tex] from [tex]\( B' \)[/tex] coordinates:
- Set [tex]\( \frac{8}{x_2} = \frac{12}{y_2} \rightarrow 8y_2 = 12x_2 \rightarrow y_2 = \frac{3}{2} x_2 \)[/tex].
6. Determine [tex]\( k \)[/tex]:
- Use [tex]\( k \)[/tex] in terms of [tex]\( x_2 \)[/tex]:
[tex]\[
k = \frac{8}{x_2}
\][/tex]
- Confirm consistency with [tex]\( y_2 \)[/tex]:
[tex]\[
y_2 = \frac{3}{2}x_2 \Rightarrow k = \frac{12}{y_2} = \frac{12}{\frac{3}{2}x_2} = \frac{12}{1.5x_2 }=8/x_2.
\][/tex]
Therefore, [tex]\( k = 2 \)[/tex].
Thus, the scale factor [tex]\( k \)[/tex] by which [tex]\(\overline{AB}\)[/tex] was dilated is:
[tex]\[
\boxed{2}
\][/tex]
