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This is a basic question, and based on my thoughts I really think this is true, but I just want some verification. If I have similar triangles, are the ratios between two sides of one triangle equal to the ratio of the corresponding sides on the other triangle?

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2 Answers

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The scaling operation that defines similarity scales all sides by the same factor. So when going from one triangle to the other, $\frac{\text{side A}}{\text{side B}}$ becomes$\frac{k×\text{side A}}{k×\text{side B}}$ and the two instances of $k$ cancel. You are right.

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In geometry, we say that two objects are similar if they have the same shape; more precisely, if one shape can be turned into the other by a combination of enlargement, rotation, and reflection. None of these transformations alter the ratios between the lengths of the sides of the triangles. Hence, two triangles can only be similar if they have the same ratios of lengths. And if these ratios are the same, then the two triangles are similar.

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