To find the magnitude of dilation from polygon ABCD to polygon A′B′C′D′ with respect to the origin, we can use the coordinates of corresponding points before and after the dilation.
The formula for dilation is given by:
\[ A' = k \cdot A \]
where \( A \) is a point in the original polygon, \( A' \) is the corresponding point in the dilated polygon, and \( k \) is the magnitude of dilation.
Let's analyze the pairs of points:
-
For point A and A':
- A(-1, -1) goes to A'(-2, -2).
- The dilation factor \( k \) can be calculated as: \[ k = \frac{x'}{x} = \frac{-2}{-1} = 2 \quad \text{and} \quad k = \frac{y'}{y} = \frac{-2}{-1} = 2. \]
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For point B and B':
- B(-1, 1) goes to B'(-2, 2).
- The dilation factor \( k \) is: \[ k = \frac{-2}{-1} = 2 \quad \text{and} \quad k = \frac{2}{1} = 2. \]
-
For point C and C':
- C(2, 3) goes to C'(4, 6).
- The dilation factor \( k \) is: \[ k = \frac{4}{2} = 2 \quad \text{and} \quad k = \frac{6}{3} = 2. \]
-
For point D and D':
- D(2, -1) goes to D'(4, -2).
- The dilation factor \( k \) is: \[ k = \frac{4}{2} = 2 \quad \text{and} \quad k = \frac{-2}{-1} = 2. \]
In each case, we consistently find that the magnitude of dilation \( k \) is 2.
Thus, the magnitude of dilation is:
\[ \text{Magnitude of dilation} = 2. \]
So the correct response is 2.
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