To determine the scale factor for the dilation of triangle ABC to triangle A'B'C', we need to analyze the relationship between the vertices of triangles ABC and A'B'C'.
The formula for dilation from a center point (in this case, the origin (0, 0)) is:
\[ (x', y') = k(x, y) \]
where \((x, y)\) are the original coordinates, \((x', y')\) are the dilated coordinates, and \(k\) is the scale factor.
We have the vertices of triangle ABC as:
- A(2, 1)
- B(-2, 2)
- C(-3, -1)
And the vertices of triangle A'B'C' as:
- A'(4, 2)
- B'(-4, 4)
- C'(-6, -2)
Now, we will find the scale factor \(k\) by comparing corresponding vertices.
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For point A (2, 1) and A' (4, 2): \[ \left( k \cdot 2, k \cdot 1 \right) = (4, 2) \] This gives: \[ k \cdot 2 = 4 \quad \Rightarrow \quad k = \frac{4}{2} = 2 \] \[ k \cdot 1 = 2 \quad \Rightarrow \quad k = \frac{2}{1} = 2 \]
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For point B (-2, 2) and B' (-4, 4): \[ \left( k \cdot (-2), k \cdot 2 \right) = (-4, 4) \] This gives: \[ k \cdot (-2) = -4 \quad \Rightarrow \quad k = \frac{-4}{-2} = 2 \] \[ k \cdot 2 = 4 \quad \Rightarrow \quad k = \frac{4}{2} = 2 \]
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For point C (-3, -1) and C' (-6, -2): \[ \left( k \cdot (-3), k \cdot (-1) \right) = (-6, -2) \] This gives: \[ k \cdot (-3) = -6 \quad \Rightarrow \quad k = \frac{-6}{-3} = 2 \] \[ k \cdot (-1) = -2 \quad \Rightarrow \quad k = \frac{-2}{-1} = 2 \]
In all cases, we find that the scale factor \(k\) is consistent.
Thus, the scale factor for the dilation of triangle ABC to triangle A'B'C' is
\[ \boxed{2} \]