Question

carl wants to prove that if a line is parallel to one side of a triangle, then it divides the other two sides proportionally.

triangle uvw with x on uv, y on uw, xy parallel to vw

select the appropriate rephrased statement for carl’s proof.
choose 1 answer:
a in △uvw, if \overleftrightarrow{xy} || \overline{vw}, then \frac{ux}{xv} = \frac{uy}{yw}.
b in △uvw, if \frac{ux}{xv} = \frac{uy}{yw}, then \overleftrightarrow{xy} || \overline{vw}.
c in △uvw, if \overleftrightarrow{xy} || \overline{vw}, then \frac{ux}{xv} = \frac{xy}{wv}.
d in △uvw, if \frac{ux}{xv} = \frac{xy}{wv}, then \overleftrightarrow{xy} || \overline{vw}.

Explanation:

The problem is about the Basic Proportionality Theorem (Thales' theorem), which states that if a line is parallel to one side of a triangle, it divides the other two sides proportionally. We need to check which option correctly rephrases this for triangle \( \triangle UVW \) with \( \overleftrightarrow{XY} \parallel \overline{VW} \).

• Option A: If \( \overleftrightarrow{XY} \parallel \overline{VW} \), then \( \frac{UX}{XV} = \frac{UY}{YW} \). This matches the theorem: the line \( XY \) (parallel to \( VW \)) divides sides \( UV \) (into \( UX \) and \( XV \)) and \( UW \) (into \( UY \) and \( YW \)) proportionally.
• Option B: This is the converse (if proportional, then parallel), but we need the direct theorem (if parallel, then proportional), so B is incorrect.
• Option C: The ratio \( \frac{UX}{XV} = \frac{XY}{WV} \) does not follow the proportionality of the divided sides; it compares a segment on \( UV \) to a segment on \( UV \) with a segment on \( XY \) to \( WV \), which is not the theorem.
• Option D: Similar to C, the ratio is incorrect and it's also the converse structure (if ratio, then parallel), which is not what we need.

So the correct option is A.

Answer:

A. In \( \triangle UVW \), if \( \overleftrightarrow{XY} \parallel \overline{VW} \), then \( \frac{UX}{XV} = \frac{UY}{YW} \).