Question

triangle efg is dilated by a scale factor of $\frac{1}{2}$ centered at (1, 1) to create triangle efg. which statement is true about the dilation? the slope of $overline{ef}$ is the same as the slope of $overline{eh}$. $overline{eg}$ will overlap $overline{eg}$. $overline{gf}$ and $overline{gf}$ both pass through the center of dilation. $overline{ef}congoverline{ef}$.

Explanation:

Step1: Recall dilation properties

Dilation is a transformation that changes the size of a figure but not its shape. The center of dilation is a fixed point.

Step2: Analyze each option

• Option A: Slope of $\overline{EF}$ and $\overline{E'H'}$ (typo in option likely meant $\overline{E'F'}$). Dilation changes distances and while parallel lines have same slope, this option is wrong as it's not a general property for non - parallel segments related to dilation in this way.
• Option B: $\overline{E'G'}$ will not overlap $\overline{EG}$ because of the scale factor of $\frac{1}{2}$. The dilated figure is smaller and non - overlapping.
• Option C: In a dilation, the lines connecting corresponding points (pre - image and image) pass through the center of dilation. So, the lines containing $\overline{GF}$ and $\overline{G'F'}$ pass through the center of dilation $(1,1)$. This is a correct property of dilation.
• Option D: Since the scale factor is $\frac{1}{2}$, $\overline{EF}

ot\cong\overline{E'F'}$ as the length of $\overline{E'F'}$ is half of the length of $\overline{EF}$.

Answer:

C. $\overline{GF}$ and $\overline{G'F'}$ both pass through the center of dilation.