Question

triangle EFI is dilated by a scale factor of 1/2 with the center of dilation at point F. Then, it is reflected over line a to create AHFG. Based on these transformations, which statement is true?

Answer 1

Only statements i.1, i.2, ii.2 are true. Triangle EFI is not similar to either triangle HFG or GFH due to the reflection flipping the orientation of the corresponding sides.

**Statement i:**

• overline FH = 2 overline FE: Dilation by a factor of 1/2 implies corresponding side lengths in the dilated and original triangles are related by a factor of 1/2. So, FH = 1/2 * FE. This statement is true.
• overline FG = 2 overline FI:** Similar logic applies here. FG = 1/2 * FI. This statement is true.
• overline HG = 2 overline EI: This statement is false. While HE and FG are related by a factor of 2 due to dilation, the reflection across line a flips the orientation of triangle EFI relative to HFG. So, HG = FI, not 2 * EI.
• Triangle EFI sim triangle HFG: Since two pairs of corresponding sides are proportional (FE ~ FH and FI ~ FG) but the third pair (EI and HG) is not, triangle EFI is not similar to triangle HFG. This statement is false.

**Statement ii:**

• 2 overline FH = overline FE: This statement contradicts the previous analysis. It's not true.
• 2 overline FG = overline FI: Similar to statement i, this statement is true.
• 2 overline HG = overline EI: As mentioned earlier, this statement is false due to the reflection.
• Triangle EFI sim triangle HFG: Due to the same reasoning as in statement i, this statement is false.

**Statement iii:**

• overline FH = 2 overline FH: This statement is nonsensical and mathematically impossible. It's false.
• overline FG = 2 overline FE: Again, this contradicts the actual relation between corresponding sides due to dilation. It's false.
• overline HG = overline 2EI: As mentioned before, this statement is false due to the reflection.
• Triangle EFI sim triangle GFH: Since none of the corresponding side length ratios are equal or proportional, triangle EFI cannot be similar to triangle GFH. This statement is false.

**Statement iv:**

• 2 overline FH = overline FI:* This statement contradicts the previous analysis. It's not true.
• 2 overline FG = overline FE: Similar to statement ii, this statement is true.
• 2 overline HG = overline EI: As mentioned earlier, this statement is false due to the reflection.
• Triangle EFI sim triangle GFH: Due to the same reasoning as in statement iii, this statement is false.