Question

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

(there is a triangle $gij$ with a line segment $hk$ parallel to $ij$, where $gj$ is split into $jk = 24$ and $kg = 18$, and $gh = 30$. we need to find $gi$.)

$gi = \square$

Explanation:

Step1: Identify Similar Triangles

Since \( \overline{IJ} \parallel \overline{HK} \), by the Basic Proportionality Theorem (Thales' theorem), \( \triangle GHK \sim \triangle GIJ \) (AA similarity, as corresponding angles are equal due to parallel lines).

Step2: Set Up Proportion

For similar triangles, the ratios of corresponding sides are equal. So, \( \frac{GH}{GI} = \frac{GK}{GJ} \).
First, find \( GJ \): \( GJ = GK + KJ = 18 + 24 = 42 \).
Let \( GI = x \). Then \( GH = x - 30 \) (since \( GI = GH + HI \), so \( GH = GI - HI = x - 30 \)). Wait, actually, \( GH = 30 \), \( GI = x \), \( GK = 18 \), \( GJ = 42 \). So the proportion is \( \frac{30}{x} = \frac{18}{42} \)? No, wait: \( \triangle GHK \sim \triangle GIJ \), so corresponding sides: \( GH \) corresponds to \( GI \), \( GK \) corresponds to \( GJ \). So \( \frac{GH}{GI} = \frac{GK}{GJ} \) → \( \frac{30}{x} = \frac{18}{42} \)? Wait, no, \( GK = 18 \), \( GJ = 18 + 24 = 42 \), \( GH = 30 \), \( GI = x \). Wait, actually, \( HI \) is part of \( GI \), so \( GI = GH + HI \)? No, the diagram: \( G \) to \( H \) is 30, \( H \) to \( I \) is some length, \( G \) to \( K \) is 18, \( K \) to \( J \) is 24. So \( GJ = 18 + 24 = 42 \). Since \( HK \parallel IJ \), \( \triangle GHK \sim \triangle GIJ \), so \( \frac{GH}{GI} = \frac{GK}{GJ} \). So \( \frac{30}{GI} = \frac{18}{42} \)? Wait, no, \( GH = 30 \), \( GI \) is the whole side, \( GK = 18 \), \( GJ = 42 \). So solving \( \frac{30}{GI} = \frac{18}{42} \) → \( 18 \cdot GI = 30 \cdot 42 \) → \( GI = \frac{30 \cdot 42}{18} \).

Step3: Calculate \( GI \)

Simplify \( \frac{30 \cdot 42}{18} \):
\( 30 \div 6 = 5 \), \( 42 \div 6 = 7 \), \( 18 \div 6 = 3 \). Wait, \( 30 \times 42 = 1260 \), \( 1260 \div 18 = 70 \). So \( GI = 70 \).

Answer:

70