To find the points \( A' \) and \( B' \) after a dilation from the origin using a scale factor of 12, we need to first determine the coordinates of points \( A \) and \( B \) before the dilation.
Let's denote the coordinates of point \( A \) as \( (x_A, y_A) \) and point \( B \) as \( (x_B, y_B) \). The formula for dilation from the origin (0,0) with a scale factor \( k \) is:
\[ (x', y') = (kx, ky) \]
Given a scale factor of 12, we apply this to both points \( A \) and \( B \):
\[ A' = (12 \cdot x_A, 12 \cdot y_A) \] \[ B' = (12 \cdot x_B, 12 \cdot y_B) \]
After finding \( A' \) and \( B' \), we can check the lengths of \( A'B' \) and \( AB \). The length of a segment between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ \text{Length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Assuming we have point coordinates from the image or problem statement (let's say, for example, \( A(1, 2) \) and \( B(3, 3) \)), we can calculate the lengths before and after dilation.
However, without the specific coordinates of \( A \) and \( B \), we cannot compute exact results for this question.
Using the provided responses, we can derive the answers based on the assumption that you might have the initial coordinates for points somewhere else.
To validate any response, the coordinates \( A' \) and \( B' \) obtained via dilation should reflect the length ratio you are aiming to prove—a \( \frac{1}{2} \) length relationship.
If you have the coordinates for \( A \) and \( B \), please provide them for an accurate calculation. Alternatively, we seem to be analyzing which option aligns with a scale factor transformation, implying the need to check proportionality correctly with respect to the responses given.
If none of the provided coordinates reflect a half-length or proportionality on the scale based on the original endpoints, a reevaluation of the coordinates initially given or requested is essential. Please clarify, and we can proceed accurately!