Question

triangle 1 is composed of line segments \\( \overline { a b }, \overline { b c } \\), and \\( \overline { c a } \\).
triangle 2 is composed of line segments \\( \overline { d e }, \overline { e f } \\), and \\( \overline { f d } \\).
the triangles are similar with a scale of \\( 5 : 2 \\).
select the true statement about the proportions of the sides.
\\( \frac { a } { e } = 1 \\)
\\( \frac { m \overline { a c } } { m \overline { d f } } = \frac { 5 } { 2 } \\)
\\( \frac { m \overline { a b } } { m \overline { d e } } = \frac { 2 } { 5 } \\)
\\( \frac { c } { f } = \frac { 5 } { 2 } \\)

Explanation:

Step1: Recall the property of similar triangles

For similar triangles, the ratio of corresponding side - lengths is equal to the scale factor. If the scale factor of triangle \(1\) to triangle \(2\) is \(k=\frac{5}{2}\), then for corresponding sides \(s_1\) (in triangle \(1\)) and \(s_2\) (in triangle \(2\)), \(\frac{s_1}{s_2}=\frac{5}{2}\)

Step2: Analyze each option

• Option \(\frac{a}{e}=1\): \(a\) and \(e\) are angles. For similar triangles, corresponding angles are equal (\(a = d\), \(b = f\), \(c=e\)), not \(\frac{a}{e}=1\) (this is a wrong - type comparison as we are asked about side - length proportions).
• Option \(\frac{m\overline{AC}}{m\overline{DF}}=\frac{5}{2}\): \(\overline{AC}\) and \(\overline{DF}\) are corresponding sides. Since the scale factor of triangle \(1\) (with sides \(\overline{AB},\overline{BC},\overline{CA}\)) to triangle \(2\) (with sides \(\overline{DE},\overline{EF},\overline{FD}\)) is \(5:2\), and \(\overline{AC}\) corresponds to \(\overline{DF}\), by the property of similar - triangle side - length ratios \(\frac{m\overline{AC}}{m\overline{DF}}=\frac{5}{2}\)
• Option \(\frac{m\overline{AB}}{m\overline{DE}}=\frac{2}{5}\): Since the scale factor of triangle \(1\) to triangle \(2\) is \(\frac{5}{2}\), for corresponding sides \(\overline{AB}\) and \(\overline{DE}\), \(\frac{m\overline{AB}}{m\overline{DE}}=\frac{5}{2}\), not \(\frac{2}{5}\)
• Option \(\frac{c}{f}=\frac{5}{2}\): \(c\) and \(f\) are angles. For similar triangles, corresponding angles are equal (\(a = d\), \(b = f\), \(c = e\)), not \(\frac{c}{f}=\frac{5}{2}\) (this is a wrong - type comparison as we are asked about side - length proportions)

Answer:

\(\frac{m\overline{AC}}{m\overline{DF}}=\frac{5}{2}\)