Answer:
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 ...
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.
