Let's solve this step-by-step.
Step 1: Understanding Dilation
Dilation is a transformation that scales a figure by a certain scale factor relative to a center of dilation, which in this case is the origin (0, 0). Given a scale factor of [tex]\( \frac{1}{2} \)[/tex], each coordinate of a point will be multiplied by [tex]\( \frac{1}{2} \)[/tex].
Step 2: Applying Dilation to Points [tex]\( A \)[/tex] and [tex]\( B \)[/tex]
Given points:
- [tex]\( A = (0, 4) \)[/tex]
- [tex]\( B = (2, 0) \)[/tex]
We need to find the new locations [tex]\( A^{\prime} \)[/tex] and [tex]\( B^{\prime} \)[/tex] after dilation.
Step 3: Calculating the Coordinates of [tex]\( A^{\prime} \)[/tex]
For point [tex]\( A = (0, 4) \)[/tex]:
[tex]\[ A^{\prime} = \left( 0 \cdot \frac{1}{2}, 4 \cdot \frac{1}{2} \right) = (0.0, 2.0) \][/tex]
Step 4: Calculating the Coordinates of [tex]\( B^{\prime} \)[/tex]
For point [tex]\( B = (2, 0) \)[/tex]:
[tex]\[ B^{\prime} = \left( 2 \cdot \frac{1}{2}, 0 \cdot \frac{1}{2} \right) = (1.0, 0.0) \][/tex]
Step 5: Determining the Relationship Between Lines [tex]\( f \)[/tex] and [tex]\( f^{\prime} \)[/tex]
When a line is dilated by a non-zero scale factor, the new line [tex]\( f^{\prime} \)[/tex] will be parallel to the original line [tex]\( f \)[/tex]. This is because dilation preserves the relative angles and distances between points, meaning the direction of the line remains unchanged, only the lengths along the line are scaled by the factor.
Conclusion
The locations of points [tex]\( A^{\prime} \)[/tex] and [tex]\( B^{\prime} \)[/tex] are:
- [tex]\( A^{\prime} = (0.0, 2.0) \)[/tex]
- [tex]\( B^{\prime} = (1.0, 0.0) \)[/tex]
The relationship between lines [tex]\( f \)[/tex] and [tex]\( f^{\prime} \)[/tex] is that they are parallel.
So, the correct detailed answer is:
The locations of [tex]\( A^{\prime} \)[/tex] and [tex]\( B^{\prime} \)[/tex] are [tex]\( A^{\prime}(0,2) \)[/tex] and [tex]\( B^{\prime}(1,0) \)[/tex]. Lines [tex]\( f \)[/tex] and [tex]\( f^{\prime} \)[/tex] are parallel.
