To solve this problem, we need to start with the points A and B that define the line segment \( AB \), perform a dilation on these points using the specified scale factor of 12 and the center of dilation at the origin, and then find the new points \( A' \) and \( B' \).
The dilation formula for a point \( (x, y) \) with respect to the origin and a scale factor \( k \) is given by:
\[ (x', y') = (kx, ky) \]
Here, we are dilating the segment \( AB \) using a scale factor of 12.
Let's denote the original points as:
- \( A(x_A, y_A) \)
- \( B(x_B, y_B) \)
Then, after dilation, the new points will be:
- \( A'(x'_A, y'_A) = (12x_A, 12y_A) \)
- \( B'(x'_B, y'_B) = (12x_B, 12y_B) \)
We are tasked with proving that the length of segment \( A'B' \) is half the length of segment \( AB \).
- Find the length of segment \( AB \): The length of the segment \( AB \) can be calculated using the distance formula:
\[ \text{Length of } AB = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} \]
- Length of segment \( A'B' \): Similarly, the length of segment \( A'B' \) is:
\[ \text{Length of } A'B' = \sqrt{(x'_B - x'_A)^2 + (y'_B - y'_A)^2} \]
Substituting the dilated points into the formula gives:
\[ \text{Length of } A'B' = \sqrt{(12x_B - 12x_A)^2 + (12y_B - 12y_A)^2} \]
- Factor out the scale factor:
\[ \text{Length of } A'B' = \sqrt{12^2 \cdot ((x_B - x_A)^2 + (y_B - y_A)^2)} = 12 \cdot \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} \]
This means:
\[ \text{Length of } A'B' = 12 \cdot \text{Length of } AB \]
- Conclusion: For \( A'B' \) to be half the length of \( AB \), we made an error in the understanding, as our computations suggest it is 12 times longer instead of half.
So, it appears the assertion that \( A'B' \) is half the length of \( AB \) might still be provided by a conceptual error in stating the problem, or perhaps there was a misunderstanding in scale factors.
In context, if the scale factor was supposed to be \( \frac{1}{2} \), then indeed \( A'B' \) would be half of \( AB \). However, this analysis shows that with a scale factor of 12, \( A'B' \) becomes twelve times longer than the original segment \( AB \).
To summarize:
- If \( A'B' \) is indicated to be half the length, there may be an error in the interpretation. The answers provided do not substantiate this without the original coordinates of \( A \) or \( B \) specified. We must also check for potential misconceptions or adjustments to the dilation parameters.
If it simply adheres to finding the correct \( A' \) and \( B' \) per the dilation process specified, the original points should be calculated since none fits the criteria given typical dilation uses perspective root according to current dimensionality and orientation.
Thus, without the original points A and B, a direct answer cannot be provided. Please verify those inputs or add correct parameters for accurate computation.
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