Question

$\overline{ij} \parallel \overline{hk}$. find $gj$.

Explanation:

Step1: Identify Similar Triangles

Since \(\overline{IH} \parallel \overline{HK}\) (wait, actually \(\overline{ID} \parallel \overline{HK}\)? Wait, the diagram shows lines with transversals, so by the Basic Proportionality Theorem (Thales' theorem), triangles \(IGJ\) and \(IHK\) are similar. So the ratios of corresponding sides are equal. Let \(GJ = x\), then \(GK = 15\), so \(KJ = x - 15\)? Wait, no, \(GJ\) is the side from \(G\) to \(J\), and \(GK = 15\), so \(GJ = GK + KJ = 15 + KJ\)? Wait, no, looking at the diagram: \(IG = 77\), \(IH = 44\), \(GK = 15\), and we need to find \(GJ\). Since \(\overline{HK} \parallel \overline{IJ}\) (assuming, because the lines are parallel, so by the Basic Proportionality Theorem, \(\frac{IH}{IG}=\frac{IK}{GJ}\)? Wait, no, let's correct: the two triangles are similar, so \(\frac{IH}{IG}=\frac{GK}{GJ}\)? Wait, no, let's define the similar triangles. Let's say \(\triangle IHK \sim \triangle IGJ\) because \(HK \parallel IJ\) (since the given lines are parallel, so corresponding angles are equal, hence similarity by AA criterion). So the ratio of sides \(IH/IG = HK/GJ\)? Wait, no, \(IH = 44\), \(IG = 77\), \(GK = 15\), and \(GJ\) is the side we need. Wait, actually, the sides \(IH\) and \(IG\) are parts of the same line, so \(IH = 44\), \(HG = IG - IH = 77 - 44 = 33\). Then, by the Basic Proportionality Theorem (Thales' theorem), \(\frac{HG}{IG}=\frac{GK}{GJ}\)? No, Thales' theorem states that if a line is drawn parallel to one side of a triangle, intersecting the other two sides, then it divides those sides proportionally. So in \(\triangle IGJ\), line \(HK\) is parallel to \(IJ\), intersecting \(IG\) at \(H\) and \(GJ\) at \(K\). So \(\frac{IH}{IG}=\frac{IK}{GJ}\)? Wait, no, \(IH\) is from \(I\) to \(H\), \(IG\) is from \(I\) to \(G\), so \(IH = 44\), \(IG = 77\), so the ratio of \(IH\) to \(IG\) is \(44/77 = 4/7\). Then, the ratio of \(GK\) to \(GJ\) should be equal? Wait, no, \(GK\) is from \(G\) to \(K\), and \(GJ\) is from \(G\) to \(J\), so \(GK = 15\), let \(GJ = x\), then \(KJ = x - 15\). Wait, no, the correct proportion is \(\frac{IH}{IG}=\frac{KJ}{GJ}\)? No, I think I messed up. Let's use the similarity ratio. Since \(\triangle IHK \sim \triangle IGJ\), the ratio of corresponding sides is \(\frac{IH}{IG}=\frac{HK}{IJ}\), but we have \(GK = 15\), which is part of \(GJ\). Wait, maybe the correct proportion is \(\frac{IH}{HG}=\frac{IK}{GK}\)? No, let's start over.

Given: \(IH = 44\), \(IG = 77\), \(GK = 15\), find \(GJ\).

Since \(HK \parallel IJ\), \(\triangle IHK \sim \triangle IGJ\) (AA similarity, as \(\angle I\) is common, and \(\angle IHK = \angle IGJ\) because \(HK \parallel IJ\), corresponding angles equal).

Therefore, the ratio of corresponding sides: \(\frac{IH}{IG}=\frac{GK}{GJ}\)? Wait, no, \(IH\) corresponds to \(IG\), and \(GK\) corresponds to \(GJ\)? No, \(IH\) is a side of the smaller triangle, \(IG\) is the corresponding side of the larger triangle. \(GK\) is a side of the smaller triangle? No, \(GK = 15\) is part of \(GJ\). Wait, maybe the sides are \(IH = 44\), \(IG = 77\), so the ratio of similarity is \(44/77 = 4/7\). Then, the side \(GK\) in the smaller triangle corresponds to \(GJ\) in the larger triangle? No, \(GK\) is 15, but that's not right. Wait, maybe \(GK\) is 15, and \(GJ\) is the side we need, so the ratio of \(IH\) to \(IG\) is equal to the ratio of \(GK\) to \(GJ\)? Wait, no, that would be \(\frac{44}{77}=\frac{15}{GJ}\), but that would give \(GJ = (77 \times 15)/44 = (7 \times 15)/4 = 105/4 = 26.25\), which is not correct. Wait, I think I mixed up the sides. Let's define the similar triangles correctly. The two triangles are \(\triangle IHK\) and \(\triangle IGJ\), with \(HK \parallel IJ\). So the sides: \(IH = 44\), \(IG = 77\), \(HK\) is parallel to \(IJ\), so the ratio of \(IH\) to \(IG\) is equal to the ratio of \(HK\) to \(IJ\), but we have \(GK = 15\), which is part of \(GJ\). Wait, maybe the correct proportion is \(\frac{IH}{HG}=\frac{IK}{GK}\), where \(HG = IG - IH = 77 - 44 = 33\). Then \(\frac{44}{33}=\frac{IK}{15}\), so \(IK = (44 \times 15)/33 = 20\), but that's not helpful. Wait, no, Thales' theorem: in \(\triangle IGJ\), line \(HK\) is parallel to \(IJ\), so \(\frac{IH}{IG}=\frac{GK}{GJ}\)? No, Thales' theorem says \(\frac{IH}{HG}=\frac{GK}{KJ}\), where \(HG = IG - IH = 33\), \(GK = 15\), \(KJ = GJ - GK = GJ - 15\). So \(\frac{44}{33}=\frac{15}{GJ - 15}\). Simplify \(44/33 = 4/3\), so \(4/3 = 15/(GJ - 15)\). Cross-multiplying: \(4(GJ - 15) = 45\), \(4GJ - 60 = 45\), \(4GJ = 105\), \(GJ = 105/4 = 26.25\)? No, that can't be. Wait, I think I got the ratio reversed. Thales' theorem: the line parallel to the base divides the other two sides proportionally. So in \(\triangle IGJ\), with base \(IJ\), and line \(HK\) parallel to \(IJ\), intersecting \(IG\) at \(H\) and \(GJ\) at \(K\), then \(\frac{IH}{IG}=\frac{IK}{GJ}\)? No, \(IH\) is from \(I\) to \(H\), \(IG\) is from \(I\) to \(G\), so \(IH/IG = HK/IJ\), but we have \(GK = 15\), which is part of \(GJ\). Wait, maybe the triangles are similar with ratio \(44/77 = 4/7\), so the side \(GK = 15\) corresponds to \(GJ\) scaled by \(4/7\)? No, that would mean \(15 = (4/7)GJ\), so \(GJ = 15 \times 7/4 = 26.25\), same as before. But that seems odd. Wait, maybe the diagram is different: maybe \(IG = 77\), \(IH = 44\), \(GK = 15\), and we need to find \(GJ\) such that \(HK \parallel IJ\), so by the Basic Proportionality Theorem, \(\frac{IH}{IG}=\frac{GK}{GJ}\)? Wait, no, that would be \(\frac{44}{77}=\frac{15}{GJ}\), so \(GJ = (77 \times 15)/44 = (7 \times 15)/4 = 105/4 = 26.25\), but that's 26.25, which is 105/4. But maybe I made a mistake in the ratio. Wait, actually, the correct ratio is \(\frac{IH}{IG}=\frac{GK}{GJ}\) is wrong. The correct ratio is \(\frac{IH}{HG}=\frac{GK}{KJ}\), where \(HG = IG - IH = 33\), \(KJ = GJ - GK = GJ - 15\). So \(\frac{44}{33}=\frac{15}{GJ - 15}\). Simplify \(44/33 = 4/3\), so \(4/3 = 15/(GJ - 15)\). Cross-multiplying: \(4(GJ - 15) = 45\) → \(4GJ = 60 + 45 = 105\) → \(GJ = 105/4 = 26.25\). Wait, but that's a fraction. Alternatively, maybe the ratio is \(\frac{IG}{IH}=\frac{GJ}{GK}\), so \(\frac{77}{44}=\frac{GJ}{15}\), which simplifies to \(\frac{7}{4}=\frac{GJ}{15}\), so \(GJ = (7 \times 15)/4 = 105/4 = 26.25\). Yes, that makes sense. Because the larger triangle has side \(IG = 77\), smaller has \(IH = 44\), so the ratio of similarity is \(77/44 = 7/4\), so the corresponding side \(GJ\) (in larger triangle) to \(GK\) (in smaller triangle? No, wait, \(GK\) is part of \(GJ\). Wait, no, if \(\triangle IHK \sim \triangle IGJ\), then the ratio of \(IG/IH = GJ/GK\), because \(IG\) corresponds to \(IH\), and \(GJ\) corresponds to \(GK\). So \(77/44 = GJ/15\), so \(GJ = (77 \times 15)/44 = (7 \times 15)/4 = 105/4 = 26.25\). So that's the answer.

Step2: Calculate GJ

Using the proportion from similar triangles: \(\frac{IG}{IH} = \frac{GJ}{GK}\)

Substitute \(IG = 77\), \(IH = 44\), \(GK = 15\):

\(\frac{77}{44} = \frac{GJ}{15}\)

Simplify \(\frac{77}{44} = \frac{7}{4}\):

\(\frac{7}{4} = \frac{GJ}{15}\)

Solve for \(GJ\):

\(GJ = \frac{7 \times 15}{4} = \frac{105}{4} = 26.25\)

Answer:

\(26.25\) (or \(\frac{105}{4}\))