Question

question 10 (multiple choice worth 1 points)
△efi is dilated by a scale factor of \\( \frac{1}{3} \\) with the center of dilation at point f. then, it is reflected over line a to create △hfg. based on these transformations, which statement is true?
image of two triangles and a line, with options below
options (paraphrased):

1. \\( \overline{fe} = \frac{fh}{3} \\), \\( \overline{fi} = \frac{fg}{3} \\), and \\( \overline{ei} = \frac{hg}{3} \\), △efi ~ △hfg
2. \\( \overline{fe} = \frac{fg}{3} \\), \\( \overline{fi} = \frac{fh}{3} \\), and \\( \overline{ei} = \frac{hg}{3} \\), △efi ~ △gfh
3. \\( \overline{fe} = \frac{fg}{3} \\), \\( \overline{fi} = \frac{fh}{3} \\), and \\( \overline{ei} = \frac{hg}{3} \\), △efi ~ △gfh (duplicate?)
4. \\( \overline{fe} = \frac{fh}{3} \\), \\( \overline{fi} = \frac{fg}{3} \\), and \\( \overline{ei} = \frac{hg}{3} \\), △efi ~ △hfg (duplicate?)

(note: exact option text may vary due to ocr, but includes proportionality and similarity statements.)

Explanation:

Step1: Properties of Dilation and Reflection

When \(\triangle EFI\) is dilated by a scale factor of \(\frac{1}{3}\) with center at \(F\), we have the following relationships for corresponding sides. If two triangles are similar (dilation creates similar triangles), for similar triangles \(\triangle EFI\) and \(\triangle HFG\) (after reflection), the ratios of corresponding sides are equal to the scale factor or its reciprocal.
Since the scale factor of dilation is \(\frac{1}{3}\), we know that \(\frac{FE}{FH}=\frac{FI}{FG}=\frac{EI}{HG}=\frac{1}{3}\). By cross - multiplying, we get \(FE=\frac{1}{3}FH\), \(FI = \frac{1}{3}FG\), and \(EI=\frac{1}{3}HG\), which can be rewritten as \(3FE = FH\), \(3FI=FG\), and \(3EI = HG\).
For similar triangles \(\triangle EFI\sim\triangle HFG\) (dilation and reflection preserve similarity), the ratio of corresponding sides gives \(\frac{FE}{FH}=\frac{FI}{FG}=\frac{EI}{HG}\).

Step2: Analyze the Ratios

We know that for similar triangles \(\triangle EFI\) and \(\triangle HFG\), the ratio of corresponding sides:

• \(\frac{FE}{FH}=\frac{1}{3}\Rightarrow FE=\frac{1}{3}FH\), \(\frac{FI}{FG}=\frac{1}{3}\Rightarrow FI=\frac{1}{3}FG\), \(\frac{EI}{HG}=\frac{1}{3}\Rightarrow EI=\frac{1}{3}HG\)
• Cross - multiplying gives \(3FE = FH\), \(3FI = FG\), \(3EI=HG\)
• For similar triangles \(\triangle EFI\sim\triangle HFG\) (by AA similarity, vertical angles at \(F\) are equal and the angles are preserved under dilation and reflection), the ratio of sides \(\frac{FE}{FH}=\frac{FI}{FG}=\frac{EI}{HG}\)

Answer:

\(\frac{FE}{FH}=\frac{FI}{FG}=\frac{1}{3}\), and \(\frac{EI}{HG}=\frac{1}{3}\) (or \(3FE = FH\), \(3FI = FG\), \(3EI=HG\) and \(\triangle EFI\sim\triangle HFG\))