The perimeters of similar triangles are in the same ratio as the corresponding sides

Answers: always, never, sometimss

Always

Step-by-step explanation:

Suppose you have triangle ABC with side lengths a, b, c. Suppose that is similar to triangle DEF with side lengths d, e, f.

Now, let k be the ratio of corresponding sides ...

k = d/a

Because the same factor applies to all sides, we also have ...

k = e/b = f/c

That is, if we multiply by the denominators of each of these fractions, we get ...

d = a·k e = b·k f = c·k

The perimeter of ΔABC is ...

perimeter (ABC) = a + b + c

The perimeter of ΔDEF is ...

perimeter (DEF) = d + e + f = a·k + b·k + c·k

perimeter (DEF) = k (a + b + c) = k·perimeter (ABC)

k = perimeter (DEF) / perimeter (ABC)

That is, the perimeters are in the same ratio as corresponding sides.