Question

using the side-splitter theorem, which segment length would complete the proportion?\\(\frac{gh}{he} = \frac{?}{jf}\\)\
\\(\bigcirc\\) gf\
\\(\bigcirc\\) jh\
\\(\bigcirc\\) gj\
\\(\bigcirc\\) ef

Explanation:

Step1: Recall Side-Splitter Theorem

The theorem states: If a line is parallel to one side of a triangle and intersects the other two sides, it divides those sides proportionally.

Step2: Identify Parallel Segments

From the diagram, \(EF \parallel GF\) is incorrect; correct parallelism is \(EF \parallel GJ\)? No, recheck: The segments \(EF\) and \(GJ\) are marked parallel. The transversal line intersects \(EG\) at \(H\) and \(FJ\) at \(J\)? Wait, the proportion given is \(\frac{GH}{HE} = \frac{?}{JF}\). The Side-Splitter Theorem applies to triangle \(EGF\) with line \(HJ\) parallel to \(EF\)? No, the correct correspondence: In triangle \(EGF\), if \(HJ \parallel EF\), then \(\frac{GH}{HE} = \frac{GJ}{JF}\).

Step3: Match Proportion

The right-hand side denominator is \(JF\), so the numerator should be \(GJ\) to satisfy the theorem.

Answer:

GJ