Question

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which statement about \\( \triangle a b c \\) and \\( \triangle d e f \\) is true?
they are similar because \\( \overline{f d} \\) is twice as long as \\( \overline{c a}, \overline{d e} \\) is twice as long as \\( \overline{a b} \\), and \\( \overline{e f} \\) is twice as long as \\( \overline{b c} \\).
they are not similar because corresponding sides are not proportional.
they are not similar because \\( \overline{f d} \\) is 6 more than \\( \overline{c a} \\), while \\( \overline{d e} \\) is 5 more than \\( \overline{a b} \\).
they are congruent.

Explanation:

Step1: Calculate the ratios of corresponding sides

For \(\triangle ABC\) and \(\triangle DEF\), assume the correspondence \(A - D\), \(B - E\), \(C - F\).
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 \(\frac{EF}{BC}=\frac{16}{8} = 2\).

Answer:

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