Question

which of the following statements is true only if triangles efi and gfh are similar?
options:
2\overline{fi} = 3\overline{fh}
lines ei and hg are parallel
points e, f, and g are collinear
\frac{\overline{ei}}{\overline{fi}} = \frac{\overline{gh}}{\overline{fh}}

Explanation:

Step1: Recall the property of similar triangles

If two triangles \(\triangle EFI\) and \(\triangle GFH\) are similar, then the ratios of their corresponding sides are equal. That is, if \(\triangle EFI\sim\triangle GFH\), then \(\frac{EI}{FI}=\frac{GH}{FH}\) (by the Side - Angle - Side (SAS) similarity criterion or the definition of similar triangles which states that for similar triangles \(\triangle ABC\) and \(\triangle DEF\), \(\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}\)).

Step2: Analyze each option

• For the option \(2\overline{FI} = 3\overline{FH}\): There is no general similarity - based reason for this specific linear relationship between \(FI\) and \(FH\) to hold just because the triangles are similar. The ratio of sides of similar triangles depends on the scale factor of similarity which is not given as \(\frac{3}{2}\) in the problem statement.
• For the option "Lines \(EI\) and \(HG\) are parallel": Parallel lines can exist in non - similar triangle configurations. For example, we can construct non - similar triangles with two sides that are parallel.
• For the option "Points \(E\), \(F\), and \(G\) are collinear": There is no direct connection between the collinearity of \(E\), \(F\), and \(G\) and the similarity of \(\triangle EFI\) and \(\triangle GFH\). We can have non - collinear points \(E\), \(F\), \(G\) and still have similar triangles (by rotating or translating the triangles) and vice - versa.

Answer:

\(\frac{\overline{EI}}{\overline{FI}}=\frac{\overline{GH}}{\overline{FH}}\)