To prove that \( A'B' \) is half the length of \( AB \) after the dilation by a scale factor of \( \frac{1}{2} \) (since a dilation with a scale factor of 12 from the origin would make points move outward, to show that \( A' \) and \( B' \) are half the distance we actually need a scale factor of \( \frac{1}{2} \) instead), we need to find the coordinates of the points \( A' \) and \( B' \).
- First, we need the original coordinates of points \( A \) and \( B \).
- The dilation occurs at the origin with the scale factor of \( 12 \).
Let’s say:
- \( A(x_A, y_A) \)
- \( B(x_B, y_B) \)
After the dilation, the points are given by the formula:
- \( A'(x', y') = (k \cdot x_A, k \cdot y_A) \)
- \( B'(x', y') = (k \cdot x_B, k \cdot y_B) \)
For a scale factor of \( 12 \), this means:
- \( A' = (12 \cdot x_A, 12 \cdot y_A) \)
- \( B' = (12 \cdot x_B, 12 \cdot y_B) \)
Since the task states that we want to show that \( A'B' \) is half the length of \( AB \), we want to check the offered responses to see if these points correspond to coordinates that match \( A \) and \( B \) when diluted by a factor of \( \frac{1}{2} \), which would show that \( AB \) is actually longer than \( A'B' \) based on the understanding of scaling.
Let's assess the responses:
- A' is at (4,8) and B' is at (12,12).
- A' is at (1,5) and B' is at (3,6).
- A' is at (1,2) and B' is at (3,3).
- A' is at (−1,−2) and B' is at (−3,−3).
For each of these pairs, if we divide the coordinates of \( A' \) and \( B' \) by \( 12 \) to find \( A \) and \( B \):
- \( A(4/12, 8/12) = (1/3, 2/3) \) and \( B(12/12, 12/12) = (1, 1) \)
- \( A(1/12, 5/12) \) and \( B(3/12, 6/12) \)
- \( A(1/12, 2/12) \) and \( B(3/12, 3/12) \)
- \( A(-1/12, -2/12) \) and \( B(-3/12, -3/12) \)
Given we are scaling half of the original segment, we need to see which response represents points that can indeed be confirmed through a scale factor presence.
To conclude, the answer that correctly reflects the shrinking nature (an understanding of negative scale) according to the predetermined limits should show values scaled before the data is determined the size. The correct answer briefly could not be asserted without originally knowing the initial size of \( AB \), presuming the visual holds accuracy. Thus, direct determination relies on visual confirmation from original plotted dependencies not included.
That said, if we assume that points A and B are such that dilation leads accurately to a half-length outcome (after considering \( A' \) and \( B' \)), then depending on visual observation, the item presenting coordinates that reduce to effect yield of identification must surface.
Without precise original values, the only conclusive origin would be to our assumptions. With those remarks, the only pairs making sense relative to scaling must see based upon interpretations of each path whether the responses were true reflects.
However, assuming measurements yield:
- If points \( A \) and \( B \)'s length on the direct observation before forming responses and yielding true ends on corresponding ends, the outcomes dealing primarily with half must show accurately.
If working alternatively, a clear response away therefore should yield the projected nature directly instead of waiting upon incorrect presentatives. Addition confirmation would provide but an actual provenance for which segments and their measurements.
If you provide the coordinates of original points \( A \) and \( B \), I can help you confirm which option correctly reflects the dilation response based on actual segmentation.
1 answer