Question

which statement about \\( \triangle abc \\) and \\( \triangle def \\) is true?
they are similar because \\( \overline{fd} \\) is twice as long as \\( \overline{ca}, \overline{de} \\) is twice as long as \\( \overline{ab} \\), and \\( \overline{ef} \\) is twice as long as \\( \overline{bc} \\).
they are congruent.
they are not similar because \\( \overline{fd} \\) is 6 more than \\( \overline{ca} \\), while \\( \overline{de} \\) is 5 more than \\( \overline{ab} \\).
they are not similar because corresponding sides are not proportional.

Explanation:

Step1: Calculate the ratios of corresponding sides

For \(\triangle ABC\) and \(\triangle DEF\), if \(CA = 6\), \(FD=12\); \(AB = 5\), \(DE = 10\); \(BC=9\), \(EF = 18\).
The ratio of \(FD\) to \(CA\) is \(\frac{FD}{CA}=\frac{12}{6} = 2\).
The ratio of \(DE\) to \(AB\) is \(\frac{DE}{AB}=\frac{10}{5}=2\).
The ratio of \(EF\) to \(BC\) is \(\frac{EF}{BC}=\frac{18}{9} = 2\).

Step2: Check the similarity condition

By the Side - Side - Side (SSS) similarity criterion, if the ratios of the corresponding sides of two triangles are equal, then the two triangles are similar.

Answer:

They are similar because \(\overline{FD}\) is twice as long as \(\overline{CA}\), \(\overline{DE}\) is twice as long as \(\overline{AB}\), and \(\overline{EF}\) is twice as long as \(\overline{BC}\).