Question

$\overline{jk} \parallel \overline{gi}$. find $gh$.

diagram with points j, i, h on a horizontal line (ji=22, ih=11), h connected to g and k, k connected to j, gk=12, and gi parallel to jk

$gh = \square$

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Explanation:

Step1: Identify Similar Triangles

Since \(\overline{JK} \parallel \overline{GI}\), by the Basic Proportionality Theorem (Thales' theorem), \(\triangle JKH \sim \triangle IGH\) (similar triangles).

Step2: Set Up Proportion

For similar triangles, the ratios of corresponding sides are equal. So, \(\frac{JI}{IH}=\frac{KG}{GH}\)? Wait, no. Wait, \(JI = 22\), \(IH = 11\), so \(JH=JI + IH=22 + 11 = 33\). Wait, actually, the ratio of \(JI\) to \(JH\) should be equal to the ratio of \(KG\) to \(KH\)? Wait, no, let's correct. Since \(\overline{GI} \parallel \overline{JK}\), the triangles \(\triangle IGH\) and \(\triangle JKH\) are similar. So, the ratio of \(IH\) to \(JH\) is equal to the ratio of \(GH\) to \(KH\). Wait, \(IH = 11\), \(JH=JI + IH = 22+11 = 33\). Let \(GH = x\), then \(KH=GH + GK=x + 12\). So, \(\frac{IH}{JH}=\frac{GH}{KH}\)? Wait, no, similar triangles: corresponding sides. So, \(\triangle IGH \sim \triangle JKH\) (because \(GI \parallel JK\), so corresponding angles are equal). Therefore, \(\frac{IH}{JH}=\frac{GH}{KH}\). Wait, \(IH = 11\), \(JH=22 + 11 = 33\), \(GH = x\), \(KH=x + 12\). So, \(\frac{11}{33}=\frac{x}{x + 12}\). Simplify \(\frac{11}{33}=\frac{1}{3}\), so \(\frac{1}{3}=\frac{x}{x + 12}\). Cross - multiply: \(x + 12=3x\). Subtract \(x\) from both sides: \(12 = 2x\). Then \(x = 6\). Wait, but let's check again. Alternatively, the ratio of \(JI\) to \(IH\) is \(22:11 = 2:1\). So, the ratio of \(JK\) to \(GI\) should be \(2:1\)? Wait, no, maybe the other way. Wait, \(JI = 22\), \(IH = 11\), so \(JI/IH = 2\). So, the triangle \(\triangle JKG\) and \(\triangle IHG\)? No, maybe I mixed up the triangles. Wait, let's look at the segments. \(JH\) is \(JI + IH=22 + 11 = 33\), \(IH = 11\), so \(IH\) is half of \(JI\) (since \(22/11 = 2\)). So, the line \(GI\) is parallel to \(JK\), so by the Basic Proportionality Theorem, \(G\) divides \(KH\) in the same ratio as \(I\) divides \(JH\). So, \(IH/JH=GH/KH\). Wait, \(IH = 11\), \(JH = 33\), so \(11/33 = 1/3\). Let \(GH = x\), \(KH=x + 12\). So, \(x/(x + 12)=1/3\), cross - multiply: \(3x=x + 12\), \(2x = 12\), \(x = 6\). Wait, but let's check with another approach. The ratio of \(JI\) to \(IH\) is \(22:11 = 2:1\), so the ratio of \(KG\) to \(GH\) should be \(2:1\)? Wait, \(KG = 12\), so if \(KG/GH = 2/1\), then \(12/GH = 2/1\), so \(GH = 6\). Yes, that makes sense. Because if \(JI\) is twice \(IH\), then \(KG\) is twice \(GH\), so \(GH=12/2 = 6\).

Answer:

\(6\)