To determine the magnitude of dilation from triangle ABC to triangle A'B'C', we can use the coordinates of corresponding points.
The coordinates of triangle ABC are:
- A(2, 1)
- B(-1, 3)
- C(-3, -2)
The coordinates of triangle A'B'C' are:
- A'(4, 2)
- B'(-2, 6)
- C'(-6, -4)
The formula for dilation with reference to the origin is given by:
\[ (x', y') = k(x, y) \]
where \( (x, y) \) are the original coordinates, \( (x', y') \) are the dilated coordinates, and \( k \) is the magnitude of dilation.
To find \( k \), we can look at the coordinates of point A:
For point A:
- Original point A = (2, 1)
- Dilated point A' = (4, 2)
Using the dilation formula: \[ k \cdot (2, 1) = (4, 2) \]
From this, we can set up the equations:
For x-coordinate: \[ k \cdot 2 = 4 \implies k = \frac{4}{2} = 2 \]
For y-coordinate: \[ k \cdot 1 = 2 \implies k = \frac{2}{1} = 2 \]
Thus, the magnitude of dilation \( k \) is 2.
Now, we can confirm this with another point, for instance point B:
For point B:
- Original point B = (-1, 3)
- Dilated point B' = (-2, 6)
Calculating \( k \): For x-coordinate: \[ k \cdot (-1) = -2 \implies k = \frac{-2}{-1} = 2 \]
For y-coordinate: \[ k \cdot 3 = 6 \implies k = \frac{6}{3} = 2 \]
Both points confirm the same magnitude of dilation.
Thus, the magnitude of dilation is: \[ \boxed{2} \]
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