Triangles ABC and DEF are similar. The similarity ratio of to is , AB = 40, BC = 25, and AC = 35. What is the perimeter of ?

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To find the perimeter of triangle DEF, we first need to determine the lengths of its sides. Since triangles ABC and DEF are similar, the corresponding sides are proportional.

Let's denote the length of DE as x. According to the similarity ratio, the corresponding side in triangle ABC, AB, is 40. Therefore, the ratio of the lengths of the corresponding sides is:

DE / AB = x / 40

We can cross-multiply to find DE:

DE = (x / 40) * AB

Now, we need to find the lengths of the other two sides, DF and EF. We'll use the same ratio:

DF / BC = x / 40

DF = (x / 40) * BC

EF / AC = x / 40

EF = (x / 40) * AC

Now that we have the lengths of all three sides of triangle DEF, we can calculate its perimeter:

Perimeter of triangle DEF = DE + DF + EF

Substituting the values we found:

Perimeter of triangle DEF = (x / 40) * AB + (x / 40) * BC + (x / 40) * AC

Substituting the given values:

Perimeter of triangle DEF = (x / 40) * 40 + (x / 40) * 25 + (x / 40) * 35

Simplifying further:

Perimeter of triangle DEF = x + (5/8) * x + (7/8) * x

Perimeter of triangle DEF = (1 + 5/8 + 7/8) * x

Perimeter of triangle DEF = (16/8 + 5/8 + 7/8) * x

Perimeter of triangle DEF = (28/8) * x

Perimeter of triangle DEF = 7/2 * x

Therefore, the perimeter of triangle DEF is equal to (7/2) times the length of side DE (x).