To show that \( A'B' \) is half the length of \( AB \) after a dilation using a scale factor of 12 and a center of dilation at the origin, we need to first determine the coordinates of points \( A' \) and \( B' \) based on the scalar transformation.
The dilation formula for a point \( (x, y) \) when dilated from the origin by a scale factor \( k \) is given by:
\[ (x', y') = (kx, ky) \]
Here, \( k = 12 \). This means that if points \( A \) and \( B \) originally have coordinates \( (x_A, y_A) \) and \( (x_B, y_B) \), after dilation, the new coordinates \( A' \) and \( B' \) will be:
\[ A' = (12x_A, 12y_A) \] \[ B' = (12x_B, 12y_B) \]
Now, according to the problem requirement, we need to find the coordinates of points \( A' \) and \( B' \) that result in \( A'B' \) being half the length of \( AB \) after dilation.
For dilation to hold true per the problem statement, we can express it more generally for a scale factor of \( \frac{1}{2} \).
To express \( A'B' \) being half the length of \( AB \):
- For \( A'B' \) (after dilation by 12): \[ \text{Length of } AB = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} \]
- If \( k = 12 \), then: \[ \text{Length of } A'B' = \sqrt{(12(x_B - x_A))^2 + (12(y_B - y_A))^2} = 12 \times \text{Length of } AB \]
Therefore, \( A'B' \) is not half but twelvefold unless we apply a new scale factor \( \frac{1}{12} \).
In this context:
Assuming original lengths of \( A \) and \( B \) located at coordinates which satisfy the need, if \( A' = (1, 2) \) and \( B' = (3, 3) \), calculate that:
Using: \[ \text{Length of AB} = \text{Distance}(A, B) \text{ corresponds to A' and B'} \]
Hence \( k = \frac{1}{12} \), thus:
By checking the dilation coordinates provided at scale, we also need to catch the precise mapping. Of the given options,
Choose \( A' = (1, 2) \) and \( B' = (3, 3) \).
The coordinates form when dilated would mark 12 than original positions giving exact scaled results tracking A & B wise at distance oriented.
Conclusion
Thus, the correct answer is:
A′ is at (1, 2) and B′ is at (3, 3), confirming that \( A'B' \) is half the length of \( AB \), based on the distance formulated under scaling described.