Question

triangle 1 is composed of line segments \\(\overline{ab}\\), \\(\overline{bc}\\), and \\(\overline{ca}\\).
triangle 2 is composed of line segments \\(\overline{de}\\), \\(\overline{ef}\\), and \\(\overline{fd}\\).
the triangles are similar with a scale of 5:2.
select the true statement about the proportions of the sides.
options:
\\(\frac{a}{e} = 1\\)
\\(\frac{m\overline{ac}}{m\overline{df}} = \frac{5}{2}\\)
\\(\frac{m\overline{ab}}{m\overline{de}} = \frac{2}{5}\\)
\\(\frac{c}{f} = \frac{5}{2}\\)

Explanation:

Step1: Recall Similar Triangles Property

Similar triangles have corresponding sides in proportion. The scale factor is \( 5:2 \), meaning the ratio of sides of Triangle 1 to Triangle 2 is \( \frac{5}{2} \).

Step2: Analyze Each Option

• Option 1: \( \frac{a}{e}=1 \) implies \( a = e \), but angles (not sides) are equal in similar triangles, so incorrect.
• Option 2: \( \frac{m\overline{AC}}{m\overline{DF}} \). Corresponding sides: \( \overline{AC} \) (Triangle 1) and \( \overline{DF} \) (Triangle 2). Ratio should be \( \frac{5}{2} \), so this is correct.
• Option 3: \( \frac{m\overline{AB}}{m\overline{DE}}=\frac{2}{5} \). Correct ratio should be \( \frac{5}{2} \), so incorrect.
• Option 4: \( \frac{c}{f}=\frac{5}{2} \). Need to check correspondence. \( c \) is \( \overline{AC} \), \( f \) is not corresponding to \( \overline{AC} \), so incorrect.

Answer:

The option with \( \boldsymbol{\frac{m\overline{AC}}{m\overline{DF}} = \frac{5}{2}} \) (the second option among the choices)