To find the scale factor of the dilation between triangle ABC and triangle A′B′C′, we can compare the corresponding side lengths of the triangles. The sides of triangle ABC are 10, 14, and 16, and the sides of triangle A′B′C′ are 35, 49, and 56.
To get the scale factor, we can take the ratio of a corresponding side length from triangle A′B′C′ to triangle ABC. We can do this for each pair of corresponding side lengths:
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For the side of length 10 in triangle ABC and the corresponding side of length 35 in triangle A′B′C′: \[ \text{Scale Factor} = \frac{35}{10} = 3.5 \]
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For the side of length 14 in triangle ABC and the corresponding side of length 49 in triangle A′B′C′: \[ \text{Scale Factor} = \frac{49}{14} = 3.5 \]
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For the side of length 16 in triangle ABC and the corresponding side of length 56 in triangle A′B′C′: \[ \text{Scale Factor} = \frac{56}{16} = 3.5 \]
Since all corresponding side lengths yield the same scale factor, we can conclude that the scale factor of the dilation from triangle ABC to triangle A′B′C′ is
\[ \boxed{3.5} \]
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