Question

eleonore wants to prove that if a line divides two sides in a triangle proportionally, then it is parallel to the third side.

triangle abc with points e on ac, d on ab, and ed drawn

select the appropriate rephrased statement for eleonore’s proof.
choose 1 answer:

a) in △abc, if $overleftrightarrow{ed} \parallel \overline{cb}$, then $dfrac{ae}{ec} = dfrac{ad}{db}$.
b) in △abc, if $dfrac{ae}{ec} = dfrac{ad}{db}$, then $overleftrightarrow{ed} \parallel \overline{cb}$.
c) in △abc, if $overleftrightarrow{ed} \parallel \overline{cb}$, then $dfrac{ae}{ec} = dfrac{ed}{cb}$.
d) in △abc, if $dfrac{ae}{ec} = dfrac{ed}{cb}$, then $overleftrightarrow{ed} \parallel \overline{cb}$.

Explanation:

Eleonore wants to prove that if a line divides two sides of a triangle proportionally, then it is parallel to the third side. Let's analyze each option:

• Option A: This is the converse (if parallel, then proportional), but we need the statement where proportionality implies parallelism. So A is incorrect.
• Option B: The condition is \(\frac{AE}{EC}=\frac{AD}{DB}\) (dividing two sides proportionally) and the conclusion is \(\overleftrightarrow{ED}\parallel\overline{CB}\) (parallel to the third side), which matches Eleonore's goal.
• Option C: The conclusion here is a proportion of the segments \(ED\) and \(CB\), not about parallelism from proportionality of the divided sides. So C is incorrect.
• Option D: The condition is a proportion of \(AE/EC\) and \(ED/CB\), not the proportional division of the two sides of the triangle. So D is incorrect.

Answer:

B. In \(\triangle ABC\), if \(\boldsymbol{\frac{AE}{EC}=\frac{AD}{DB}}\), then \(\boldsymbol{\overleftrightarrow{ED}\parallel\overline{CB}}\)