To find the scale factor of the dilation between triangle ABC and triangle A'B'C', we can use the lengths of corresponding sides of the triangles.
First, we need to determine the lengths of the sides of triangle ABC. We can calculate the distances between the points as follows:
-
Distance AB: \[ AB = \sqrt{(3 - 2)^2 + (2 - 1)^2} = \sqrt{1^2 + 1^2} = \sqrt{2} \]
-
Distance BC: \[ BC = \sqrt{(3 - 1)^2 + (2 - 3)^2} = \sqrt{(2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5} \]
-
Distance CA: \[ CA = \sqrt{(2 - 1)^2 + (1 - 3)^2} = \sqrt{(1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} \]
Now we have the lengths of the sides of triangle ABC:
- AB = \(\sqrt{2}\)
- BC = \(\sqrt{5}\)
- CA = \(\sqrt{5}\)
Next, we calculate the lengths of the sides of triangle A'B'C':
-
Distance A'B': \[ A'B' = \sqrt{(9 - 6)^2 + (6 - 3)^2} = \sqrt{(3)^2 + (3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \]
-
Distance B'C': \[ B'C' = \sqrt{(3 - 9)^2 + (9 - 6)^2} = \sqrt{(-6)^2 + (3)^2} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5} \]
-
Distance C'A': \[ C'A' = \sqrt{(6 - 3)^2 + (3 - 9)^2} = \sqrt{(3)^2 + (-6)^2} = \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5} \]
Now we have the lengths of the sides of triangle A'B'C':
- A'B' = \(3\sqrt{2}\)
- B'C' = \(3\sqrt{5}\)
- C'A' = \(3\sqrt{5}\)
To find the scale factor (k) of the dilation from triangle ABC to triangle A'B'C', we can take the ratio of the corresponding side lengths.
Taking AB and A'B': \[ k = \frac{A'B'}{AB} = \frac{3\sqrt{2}}{\sqrt{2}} = 3 \]
We can check this with the other sides: For BC and B'C': \[ k = \frac{B'C'}{BC} = \frac{3\sqrt{5}}{\sqrt{5}} = 3 \]
For CA and C'A': \[ k = \frac{C'A'}{CA} = \frac{3\sqrt{5}}{\sqrt{5}} = 3 \]
In all cases, the scale factor is \(3\).
Thus, the scale factor of the dilation between triangle ABC and triangle A'B'C' is: \[ \boxed{3} \]