To prove that \( A'B' \) is half the length of \( AB \) after a dilation of \( AB \) with a scale factor of 12 and a center of dilation at the origin, we first need to find the new points \( A' \) and \( B' \) based on their respective original coordinates \( A(x_A, y_A) \) and \( B(x_B, y_B) \).
The formula for dilation is given by: \[ A' = k \cdot A \] \[ B' = k \cdot B \] where \( k \) is the scale factor and \( A' \) and \( B' \) are the new coordinates after dilation.
Given:
- Scale factor \( k = \frac{1}{12} \)
- Center of dilation at the origin (0, 0)
Let's assume original points \( A \) and \( B \) are \( A = (x_A, y_A) \) and \( B = (x_B, y_B) \).
After dilation by scale factor of \(\frac{1}{12}\):
\[ A' = \left( \frac{x_A}{12}, \frac{y_A}{12} \right) \] \[ B' = \left( \frac{x_B}{12}, \frac{y_B}{12} \right) \]
Next, let's calculate the length \( AB \) and \( A'B' \):
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Distance between points \( A \) and \( B \): \[ AB = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} \]
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Distance between points \( A' \) and \( B' \): \[ A'B' = \sqrt{\left( \frac{x_B}{12} - \frac{x_A}{12} \right)^2 + \left( \frac{y_B}{12} - \frac{y_A}{12} \right)^2} \] \[ A'B' = \sqrt{\frac{(x_B - x_A)^2}{144} + \frac{(y_B - y_A)^2}{144}} = \frac{1}{12}\sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} = \frac{1}{12}AB \]
Therefore, since we have a scale factor of \( \frac{1}{12} \), the conclusion is: \[ A'B' = \frac{1}{12}AB \] So to answer the question, none of the choices presented seem to give a valid definition that corresponds to this dilation context.
If we want \( A'B' \) to be half the original \( AB \), then we should use a scale factor of \( 2 \).
Therefore, we can conclude that with the given scale factor of \( \frac{1}{12} \), \( A'B' \) will not be half the length of \( AB \).
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