Question

vega wants to prove that the segment joining midpoints of two sides of a triangle is half the length of the third side.

select the appropriate rephrased statement for vegas proof.
choose 1 answer:
a in $\triangle efg$, if $fg = \frac{1}{2}hi$, then $eh = hf$ and $ei = ig$.
b in $\triangle efg$, if $hi = \frac{1}{2}fg$, then $eh = hf$ and $ei = ig$.
c in $\triangle efg$, if $eh = hf$ and $ei = ig$, then $fg = \frac{1}{2}hi$.
d in $\triangle efg$, if $eh = hf$ and $ei = ig$, then $hi = \frac{1}{2}fg$.

Explanation:

Vega wants to prove that the midsegment (HI) of triangle EFG (joining midpoints H and I, where EH = HF and EI = IG) is half the length of the third side (FG). So the correct rephrased statement should have the condition as EH = HF and EI = IG (midpoints) and the conclusion as HI = $\frac{1}{2}$FG. Option D matches this: if EH = HF and EI = IG (midpoints), then HI = $\frac{1}{2}$FG. Options A and C have the ratio reversed, and option B has the condition and conclusion mixed (it assumes HI = $\frac{1}{2}$FG to conclude midpoints, but we need to assume midpoints to conclude HI = $\frac{1}{2}$FG).

Answer:

D. In $\triangle EFG$, if $EH = HF$ and $EI = IG$, then $HI=\frac{1}{2}FG$.