To find the magnitude of dilation from parallelogram ABCD to parallelogram A'B'C'D', we need to compare the coordinates of corresponding points before and after dilation, focusing on how the coordinates change relative to the origin (0,0).
Given:
- A(-2, 4) transforms to A'(-1, 2)
- B(4, 4) transforms to B'(2, 2)
- C(2, -2) transforms to C'(1, -1)
- D(-4, -2) transforms to D'(-2, -1)
To find the dilation factor \( k \), we can use the x-coordinates or y-coordinates of any corresponding point pairs:
For A(-2, 4) to A'(-1, 2):
- The x-coordinate: \[ k = \frac{-1}{-2} = \frac{1}{2} \]
- The y-coordinate: \[ k = \frac{2}{4} = \frac{1}{2} \]
For B(4, 4) to B'(2, 2):
- The x-coordinate: \[ k = \frac{2}{4} = \frac{1}{2} \]
- The y-coordinate: \[ k = \frac{2}{4} = \frac{1}{2} \]
For C(2, -2) to C'(1, -1):
- The x-coordinate: \[ k = \frac{1}{2} \]
- The y-coordinate: \[ k = \frac{-1}{-2} = \frac{1}{2} \]
For D(-4, -2) to D'(-2, -1):
- The x-coordinate: \[ k = \frac{-2}{-4} = \frac{1}{2} \]
- The y-coordinate: \[ k = \frac{-1}{-2} = \frac{1}{2} \]
In each case, the magnitude of dilation \( k \) is \( \frac{1}{2} \).
Thus, the correct response for the magnitude of dilation is: Start Fraction 1 over 2 End Fraction, which corresponds to the proportion of \( \frac{1}{2} \) that satisfies all transformations.