To find the points [tex]\( A' \)[/tex] and [tex]\( B' \)[/tex] after a dilation with a scale factor of [tex]\(\frac{1}{2}\)[/tex] and a center of dilation at the origin, we need to follow these steps:
1. Understand the Dilation Transformation:
A dilation transformation will change each coordinate of a point by multiplying it with the scale factor. Here the scale factor is [tex]\(\frac{1}{2}\)[/tex], so each coordinate will be halved.
2. Apply the Transformation to Point [tex]\( A \)[/tex]:
Let's start with point [tex]\( A \)[/tex].
[tex]\[ A = (2, 4) \][/tex]
To find [tex]\( A' \)[/tex], we multiply each coordinate by [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[
A' = \left(2 \cdot \frac{1}{2}, 4 \cdot \frac{1}{2}\right) = (1, 2)
\][/tex]
3. Apply the Transformation to Point [tex]\( B \)[/tex]:
Now let's apply the same process to point [tex]\( B \)[/tex].
[tex]\[ B = (6, 6) \][/tex]
To find [tex]\( B' \)[/tex], we multiply each coordinate by [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[
B' = \left(6 \cdot \frac{1}{2}, 6 \cdot \frac{1}{2}\right) = (3, 3)
\][/tex]
4. Verification of Points:
Based on the process described, we have found [tex]\( A' = (1, 2) \)[/tex] and [tex]\( B' = (3, 3) \)[/tex].
5. Comparison with Given Options:
We need to check the calculated points against the given options to determine the correct choice:
- Option 1: [tex]\( A' \)[/tex] is at [tex]\( (1, 2) \)[/tex] and [tex]\( B' \)[/tex] is at [tex]\( (3, 3) \)[/tex].
- Option 2: [tex]\( A' \)[/tex] is at [tex]\( (1, 5) \)[/tex] and [tex]\( B' \)[/tex] is at [tex]\( (3, 6) \)[/tex].
- Option 3: [tex]\( A' \)[/tex] is at [tex]\( (-1, -2) \)[/tex] and [tex]\( B' \)[/tex] is at [tex]\( (-3, -3) \)[/tex].
- Option 4: [tex]\( A' \)[/tex] is at [tex]\( (4, 8) \)[/tex] and [tex]\( B' \)[/tex] is at [tex]\( (12, 12) \)[/tex].
From our calculations, it is clear that the correct choice is:
[tex]\[ \boxed{1} \][/tex]
