To determine the scale factor of the dilation from triangle ABC to triangle A′B′C′, we can calculate the distance between the origin (0, 0) to a vertex of triangle ABC and the corresponding vertex of triangle A′B′C′.
Let's consider the vertex A and A′:
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A is at (−3, −3), and its distance from the origin can be calculated as: \[ d_A = \sqrt{(-3)^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \]
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A′ is at (−9, −9), and its distance from the origin can be calculated as: \[ d_{A'} = \sqrt{(-9)^2 + (-9)^2} = \sqrt{81 + 81} = \sqrt{162} = 9\sqrt{2} \]
Now, we find the scale factor \(k\) by taking the ratio of the distances: \[ k = \frac{d_{A'}}{d_A} = \frac{9\sqrt{2}}{3\sqrt{2}} = \frac{9}{3} = 3 \]
Now let's validate this with the other vertices:
For B(3, 3) and B′(9, 9):
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Distance from the origin to B: \[ d_B = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \]
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Distance from the origin to B′: \[ d_{B'} = \sqrt{9^2 + 9^2} = \sqrt{81 + 81} = \sqrt{162} = 9\sqrt{2} \]
The scale factor: \[ k = \frac{d_{B'}}{d_B} = \frac{9\sqrt{2}}{3\sqrt{2}} = \frac{9}{3} = 3 \]
For C(0, 3) and C′(0, 9):
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Distance from the origin to C is: \[ d_C = \sqrt{0^2 + 3^2} = \sqrt{9} = 3 \]
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Distance from the origin to C′ is: \[ d_{C'} = \sqrt{0^2 + 9^2} = \sqrt{81} = 9 \]
The scale factor: \[ k = \frac{d_{C'}}{d_C} = \frac{9}{3} = 3 \]
Since we have calculated the same scale factor for all vertices, we conclude that the scale factor used for the dilation from triangle ABC to triangle A′B′C′ is: \[ \boxed{3} \]