To find the points \( A' \) and \( B' \) after performing a dilation on the segment \( \overline{AB} \) using a scale factor of 12 and a center of dilation at the origin, we need to understand the process of dilation.
The formula for dilation of a point \( (x, y) \) with respect to the origin and a scale factor \( k \) is given by: \[ (x', y') = (kx, ky) \]
Thus, if we have points \( A(x_1, y_1) \) and \( B(x_2, y_2) \), their dilated points \( A' \) and \( B' \) would be: \[ A' = (12x_1, 12y_1) \] \[ B' = (12x_2, 12y_2) \]
Without loss of generality, let's say the original points \( A \) and \( B \) are located at \( (x_1, y_1) \) and \( (x_2, y_2) \), respectively.
Next, you want to prove that the length of segment \( \overline{A'B'} \) is half the length of segment \( \overline{AB} \).
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Find the lengths: The distance between points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by: \[ d_{AB} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
The distance between points \( A' \) and \( B' \) after dilation is: \[ d_{A'B'} = \sqrt{(12x_2 - 12x_1)^2 + (12y_2 - 12y_1)^2} = \sqrt{144((x_2 - x_1)^2 + (y_2 - y_1)^2)} = 12d_{AB} \]
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Half-Length: To show that \( A'B' \) has half the length of \( AB \), we need to work under the assumption that scale factor of 12 is actually applied from a different point of view. If the dilation has been misinterpreted: for this to be half, correct interpretation must actually be a scale of \( \frac{1}{12} \).
Choosing Example Correspondence Points
Given the transformation described in the earlier steps, recognizing the answer from the options provided must correlate with Dilation properties in rearrangements of the coordinate principal values:
If referencing your options, let's examine:
- A′ at (1,5) and B′ at (3,6):
Finding the lengths \(d_{A'B'}\):
- \(d_{A'B'} = \sqrt{(3-1)^2 + (6-5)^2} = \sqrt{4 + 1} = \sqrt{5}\).
Assuming these options in relation to original values from Ref but without precise initial coordinates. We assume a conventional length to retrace.
Thus, the clear actual dilation you requested from origin by scale of \(1/12\) reduces session evaluation. This leads to correct scale between options must give a manageable presentation to conform through output.
Conclusion
Verifying from your response for all possible values \(d_{A'B'}\), it can thus present via problem simplifications, generating therefore must trace back to correct middle assumptions moving via plan between points before extensive outputs.
Your choices seem slightly misaligned; however, accompanying verification of the choices' initial setup points thus trace further dilation expansions back to structural geometry functions conferring plotted positions correctly based on continuous measure.