To find the magnitude of the dilation from parallelogram ABCD to A'B'C'D', we can analyze the relationship between the corresponding points.
Dilation is defined as the transformation of points in a plane relative to a center point (in this case, the origin) by a scale factor. Mathematically, for a point \( (x, y) \) being dilated to a point \( (x', y') \) in reference to the origin by a scale factor \( k \), this relationship can be expressed as:
\[ (x', y') = (k \cdot x, k \cdot y) \]
We can find the scale factor \( k \) by using the coordinates of corresponding points.
Let's take point A and A':
- Original point A: \( (-2, 4) \)
- Dilated point A': \( (-1, 2) \)
Using the formula:
\[ (-1, 2) = (k \cdot -2, k \cdot 4) \]
So we have the equations:
- \( k \cdot -2 = -1 \)
- \( k \cdot 4 = 2 \)
From the first equation, solving for \( k \): \[ k \cdot -2 = -1 \quad \Rightarrow \quad k = \frac{-1}{-2} = \frac{1}{2} \]
From the second equation, solving for \( k \): \[ k \cdot 4 = 2 \quad \Rightarrow \quad k = \frac{2}{4} = \frac{1}{2} \]
In both cases, we find that \( k = \frac{1}{2} \).
This indicates that the magnitude of dilation from parallelogram ABCD to A'B'C'D' is:
\[ \boxed{\frac{1}{2}} \]
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