If ∆ ABC ~ ∆ DEF, BC = 12 cm, EF = 10 cm, and AC = 18 cm, find DF.

by ratios

DF/10 = 18/12

solve for DF

The triangles are similar (but not congruent).

That means that corresponding sides are all in the same ratio.
BC/EF = 12/10 = AC/DF = 18/DF

DF = 18*(5/6) = 15

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To find the length of DF, we can use the concept of similarity and the corresponding sides of similar triangles.

Given that ∆ ABC ~ ∆ DEF, this means that the corresponding sides of the two triangles are proportional.

We can set up a similarity ratio using the corresponding sides:
AB/DE = BC/EF = AC/DF

Substituting the given values:
AB/DE = 12/10 = 18/DF

To find DF, we can cross multiply and solve for DF:
12 * DF = 10 * 18

Multiplying:
12DF = 180

Dividing both sides by 12:
DF = 180/12

Simplifying:
DF = 15 cm

Therefore, the length of DF is 15 cm.