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 Side - Splitter Theorem states that if a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. In the given triangle (implied by the diagram with parallel segments, as there are markings indicating parallel lines), we have a line (the one with points H and J) that is parallel to one side of the triangle.
Let's consider the triangle formed by points E, F, G (or the relevant triangle). The line \(HJ\) is parallel to \(EF\) (from the markings). So, by the Side - Splitter Theorem, \(\frac{GH}{HE}=\frac{GJ}{JF}\)? Wait, no, wait. Wait, let's look at the segments. The proportion is \(\frac{GH}{HE}=\frac{?}{JF}\). Let's identify the corresponding segments. The Side - Splitter Theorem: if a line parallel to \(EF\) (since \(E - H - G\) and \(F - J - \) the other side) intersects \(EG\) at \(H\) and \(FG\) (wait, no, the other side is \(FG\)? Wait, the diagram has \(E\), \(F\), \(G\) with \(H\) on \(EG\) and \(J\) on \(FG\) (or the other side). Wait, actually, the correct application: the line \(HJ\) is parallel to \(EF\), so in triangle \(EFG\) (assuming), line \(HJ\parallel EF\), so it divides \(EG\) and \(FG\) proportionally. So \(\frac{GH}{HE}=\frac{GJ}{JF}\)? No, wait, maybe the triangle is \(E - H - G\) and \(F - J - \) the line. Wait, let's re - express. The Side - Splitter Theorem: In a triangle, if a line is parallel to one side, then \(\frac{\text{segment on one side}}{\text{segment on the same side}}=\frac{\text{segment on the other side}}{\text{segment on the other side}}\). So, looking at the proportion \(\frac{GH}{HE}=\frac{?}{JF}\). So \(GH\) and \(HE\) are on side \(EG\) (with \(G - H - E\)), and \(?\) and \(JF\) are on side \(FG\) (or the other side). The corresponding segment to \(GH\) on the other side should be \(GJ\) and to \(HE\) should be \(JF\)? Wait, no, let's think again. Let's assume that \(HJ\parallel EF\). Then triangle \(G H J\) is similar to triangle \(G E F\) (by AA similarity, since \(\angle G\) is common and \(\angle GHJ=\angle GEF\) because \(HJ\parallel EF\)). So by similarity, \(\frac{GH}{GE}=\frac{GJ}{GF}\), but we need \(\frac{GH}{HE}=\frac{?}{JF}\). Let's re - arrange the segments. Let \(GE=GH + HE\) and \(GF = GJ+JF\). From the similarity \(\frac{GH}{GH + HE}=\frac{GJ}{GJ + JF}\). Cross - multiplying: \(GH(GJ + JF)=GJ(GH + HE)\). \(GH\times GJ+GH\times JF=GJ\times GH+GJ\times HE\). Subtract \(GH\times GJ\) from both sides: \(GH\times JF = GJ\times HE\). Then \(\frac{GH}{HE}=\frac{GJ}{JF}\). Wait, no, that's not. Wait, maybe I got the sides wrong. Let's look at the options. The options are \(GF\), \(JH\), \(GJ\), \(EF\). Let's check the proportion. The Side - Splitter Theorem: If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. So, in the diagram, the line \(HJ\) is parallel to \(EF\) (from the markings). So, the two sides are \(EG\) (with \(H\)) and \(FG\) (with \(J\)). So, \(\frac{GH}{HE}=\frac{GJ}{JF}\). Wait, \(GH\) is part of \(EG\) (from \(G\) to \(H\)), \(HE\) is from \(H\) to \(E\), \(GJ\) is part of \(FG\) (from \(G\) to \(J\)), and \(JF\) is from \(J\) to \(F\). So by the Side - Splitter Theorem, \(\frac{GH}{HE}=\frac{GJ}{JF}\). So the missing segment is \(GJ\).

Step2: Verify the Options

- Option \(GF\): \(GF = GJ + JF\), which is not the correct corresponding segment.
- Option \(JH\): \(JH\) is not a segment that corresponds in the proportion.
- Option \(GJ\): As per the Side - Splitter Theorem application above, \(\frac{GH}{HE}=\frac{GJ}{JF}\), so this is correct.
- Option \(EF\): \(EF\) is the side parallel to \(HJ\), not a segment in the proportion for the sides divided by the parallel line.

Answer:

GJ (the option corresponding to \(GJ\), which is one of the options, so the answer is the option with \(GJ\))