To find the magnitude of dilation that transforms triangle ABC into triangle A'B'C', we need to calculate the scale factor between corresponding points.
The dilation from point \(A\) to \(A'\) can be expressed in terms of the coordinates:
- \(A(2, 1)\) corresponds to \(A'(4, 2)\).
- The scale factor (k) can be calculated using the formula:
\[ k = \frac{\text{Distance from origin to } A'}{\text{Distance from origin to } A} \]
Calculating the distances:
-
Distance from the origin to \(A\): \[ d_A = \sqrt{(2-0)^2 + (1-0)^2} = \sqrt{4 + 1} = \sqrt{5} \]
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Distance from the origin to \(A'\): \[ d_{A'} = \sqrt{(4-0)^2 + (2-0)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \]
Now we can find the scale factor \(k\): \[ k = \frac{d_{A'}}{d_A} = \frac{2\sqrt{5}}{\sqrt{5}} = 2 \]
Now, let’s check if this scale factor works for the other points:
- For point \(B(-1, 3)\) to point \(B'(-2, 6)\):
Calculating distances:
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Distance from the origin to \(B\): \[ d_B = \sqrt{(-1-0)^2 + (3-0)^2} = \sqrt{1 + 9} = \sqrt{10} \]
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Distance from the origin to \(B'\): \[ d_{B'} = \sqrt{(-2-0)^2 + (6-0)^2} = \sqrt{4 + 36} = \sqrt{40} = 2\sqrt{10} \]
Scale factor for \(B\) and \(B'\): \[ k = \frac{d_{B'}}{d_B} = \frac{2\sqrt{10}}{\sqrt{10}} = 2 \]
- For point \(C(-3, -3)\) to point \(C'(-6, -4)\):
Calculating distances:
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Distance from the origin to \(C\): \[ d_C = \sqrt{(-3-0)^2 + (-3-0)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \]
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Distance from the origin to \(C'\): \[ d_{C'} = \sqrt{(-6-0)^2 + (-4-0)^2} = \sqrt{36 + 16} = \sqrt{52} = 2\sqrt{13} \]
Scale factor for \(C\) and \(C'\): \[ k = \frac{d_{C'}}{d_C} = \frac{2\sqrt{13}}{3\sqrt{2}} \]
In checking, we find that the scale factor confirms a consistent dilation from each original point to its image in the dilated triangle based on the corresponding distances to the origin.
In conclusion, the magnitude of the dilation used to transform triangle ABC into triangle A'B'C' is \(2\).
Therefore, triangle ABC is dilated using a magnitude of \( \text{2} \).
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