To determine the scale factor of the dilation from triangle ABC to triangle A′B′C′, we can analyze the coordinates of corresponding points from both triangles.
- Point A(−13, −4) dilates to A′(−32.5, −10).
- Point B(−6, −4) dilates to B′(−15, −10).
- Point C(−6, 4) dilates to C′(−15, 10).
To find the scale factor, we take the distance from the origin to each point before and after dilation.
For point A and A′:
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Distance from the origin to A: \[ \sqrt{(-13)^2 + (-4)^2} = \sqrt{169 + 16} = \sqrt{185} \]
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Distance from the origin to A′: \[ \sqrt{(-32.5)^2 + (-10)^2} = \sqrt{1056.25 + 100} = \sqrt{1156.25} = 34.0 \]
Now, we take the ratio of the distances to find the scale factor: \[ \text{Scale factor} = \frac{\text{Distance to A′}}{\text{Distance to A}} = \frac{34.0}{\sqrt{185}} \]
Calculating \(\sqrt{185} \approx 13.6\), so the scale factor is approximately: \[ \frac{34.0}{13.6} \approx 2.5 \]
To confirm this calculation, let's check point B and B′:
For point B(−6, −4) and B′(−15, −10):
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Distance from the origin to B: \[ \sqrt{(-6)^2 + (-4)^2} = \sqrt{36 + 16} = \sqrt{52} \]
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Distance from the origin to B′: \[ \sqrt{(-15)^2 + (-10)^2} = \sqrt{225 + 100} = \sqrt{325} \approx 18.03 \]
Calculating the scale factor again: \[ \text{Scale factor} = \frac{\sqrt{325}}{\sqrt{52}} \approx \frac{18.03}{7.21} \approx 2.5 \]
Now for point C and C′:
For point C(−6, 4) and C′(−15, 10):
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Distance from the origin to C: \[ \sqrt{(-6)^2 + 4^2} = \sqrt{36 + 16} = \sqrt{52} \]
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Distance from the origin to C′: \[ \sqrt{(-15)^2 + (10)^2} = \sqrt{225 + 100} = \sqrt{325} \approx 18.03 \]
Calculating the scale factor: \[ \text{Scale factor} = \frac{\sqrt{325}}{\sqrt{52}} \approx \frac{18.03}{7.21} \approx 2.5 \]
Since all calculations point to the same scale factor, we conclude:
The scale factor used to create the image is 2.5.