To determine if \( A'B' \) is a dilation of \( AB \), we first need to calculate the lengths of the segments \( AB \) and \( A'B' \), and then check for the scale factor.
Step 1: Finding the Length of Segment \( AB \)
Given points:
- \( A (-2, 0.5) \)
- \( B (0, -0.5) \)
The length of segment \( AB \) can be calculated using the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substituting the coordinates: \[ d_{AB} = \sqrt{(0 - (-2))^2 + (-0.5 - 0.5)^2} = \sqrt{(2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5} \]
Step 2: Finding the Length of Segment \( A'B' \)
Given points:
- \( A' (-4, 1) \)
- \( B' (0, -1) \)
Calculating the length of segment \( A'B' \): \[ d_{A'B'} = \sqrt{(0 - (-4))^2 + (-1 - 1)^2} = \sqrt{(4)^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \]
Step 3: Determine the Scale Factor
To find the scale factor \( k \), we compare the lengths of \( A'B' \) and \( AB \): \[ k = \frac{d_{A'B'}}{d_{AB}} = \frac{2\sqrt{5}}{\sqrt{5}} = 2 \]
Step 4: Determine if it is an Enlargement or a Reduction
Since the scale factor \( k = 2 \) is greater than 1, this indicates that \( A'B' \) is an enlargement of \( AB \).
Conclusion
Yes, \( A'B' \) is a dilation of \( AB \), with a scale factor of 2, which is an enlargement.
Therefore, the correct response is: Yes, A′B′ is a dilation of AB, with a scale factor of 2, which is an enlargement.
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