Question

in the drawing below, \\( \triangle abc \\) is similar to \\( \triangle def \\).
which choice shows the corresponding sides as equal ratios?
\\( \frac { a b } { d e } = \frac { e f } { b c } = \frac { d f } { a c } \\)
\\( \frac { d e } { a b } = \frac { e f } { b c } = \frac { a c } { d f } \\)
\\( \frac { d e } { a b } = \frac { e f } { b c } = \frac { d f } { a c } \\)
\\( \frac { d e } { a b } = \frac { b c } { e f } = \frac { d f } { a c } \\)

Explanation:

Step1: Identify corresponding sides

Since \(\triangle ABC\sim\triangle DEF\), \(AB\) corresponds to \(DE\), \(BC\) corresponds to \(EF\), and \(AC\) corresponds to \(DF\).

Step2: Calculate ratios

For \(AB = 3\), \(DE=6\), \(\frac{DE}{AB}=\frac{6}{3} = 2\); for \(BC = 4\), \(EF = 8\), \(\frac{EF}{BC}=\frac{8}{4}=2\); for \(AC = 5\), \(DF = 10\), \(\frac{DF}{AC}=\frac{10}{5}=2\). So \(\frac{DE}{AB}=\frac{EF}{BC}=\frac{DF}{AC}\) holds.

Answer:

\(\frac{DE}{AB}=\frac{EF}{BC}=\frac{DF}{AC}\)