To determine the scale factor of the dilation from triangle \( ABC \) to triangle \( A'B'C' \), we can use the formula for the scale factor based on the coordinates of a point before and after dilation.
From the problem, we have the following vertices for triangle \( ABC \):
- Point \( A(2, 1) \)
- Point \( B(-2, 2) \)
- Point \( C(-3, -1) \)
And the corresponding vertices for triangle \( A'B'C' \) after dilation are:
- Point \( A'(4, 2) \)
- Point \( B'(-4, 4) \)
- Point \( C'(-6, -2) \)
The scale factor \( k \) can be calculated using the formula:
\[ k = \frac{\text{distance from the origin to the dilated point}}{\text{distance from the origin to the original point}} \]
Let’s calculate it using point \( A \) and point \( A' \):
-
Distance from the origin to \( A \): \[ \text{Distance to } A = \sqrt{(2 - 0)^2 + (1 - 0)^2} = \sqrt{4 + 1} = \sqrt{5} \]
-
Distance from the origin to \( A' \): \[ \text{Distance to } A' = \sqrt{(4 - 0)^2 + (2 - 0)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \]
Now, calculate the scale factor \( k \): \[ k = \frac{\text{Distance to } A'}{\text{Distance to } A} = \frac{2\sqrt{5}}{\sqrt{5}} = 2 \]
We can check the other points using the same method:
For point \( B \) and point \( B' \):
-
Distance from the origin to \( B \): \[ \text{Distance to } B = \sqrt{(-2 - 0)^2 + (2 - 0)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]
-
Distance from the origin to \( B' \): \[ \text{Distance to } B' = \sqrt{(-4 - 0)^2 + (4 - 0)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
Calculating \( k \): \[ k = \frac{4\sqrt{2}}{2\sqrt{2}} = 2 \]
For point \( C \) and point \( C' \):
-
Distance from the origin to \( C \): \[ \text{Distance to } C = \sqrt{(-3 - 0)^2 + (-1 - 0)^2} = \sqrt{9 + 1} = \sqrt{10} \]
-
Distance from the origin to \( C' \): \[ \text{Distance to } C' = \sqrt{(-6 - 0)^2 + (-2 - 0)^2} = \sqrt{36 + 4} = \sqrt{40} = 2\sqrt{10} \]
Calculating \( k \): \[ k = \frac{2\sqrt{10}}{\sqrt{10}} = 2 \]
Thus, the scale factor for the dilation of triangle \( ABC \) with a center point of dilation at the origin is:
\[ \boxed{2} \]