Question

question 8 (multiple choice worth 1 points)
△hfg is dilated by a scale factor of 2 with the center of dilation at point f. then, it is reflected over line a to create △efi. based on these transformations, which statement is true?

Explanation:

Step1: Understand dilation property

When a figure is dilated by a scale factor of $k$, the lengths of corresponding sides of the original and dilated - figure are related by the scale - factor. Here, $k = 2$. So, if $\triangle HFG$ is dilated by a scale factor of 2 with center of dilation at $F$ to get a pre - reflected figure of $\triangle EFI$, the lengths of the sides of $\triangle HFG$ are half of the corresponding sides of $\triangle EFI$. That is, if we consider corresponding sides, $FG=\frac{1}{2}FI$, $FH=\frac{1}{2}FE$, and $HG=\frac{1}{2}EI$.

Step2: Understand reflection property

A reflection is a rigid transformation that preserves the shape and size of the figure. After reflection of the dilated figure over line $a$ to get $\triangle EFI$, the similarity of the triangles formed by dilation is still maintained. Since dilation and reflection are both transformations that preserve angle measures, $\triangle HFG\sim\triangle EFI$.

Answer:

The first option (where $FG = \frac{1}{2}FI$, $FH=\frac{1}{2}FE$, and $HG=\frac{1}{2}EI$ and $\triangle HFG\sim\triangle EFI$) is true.