Question

find ij.

triangle diagram with points g, h, i, k, j; gh = 42, gi = 63, kj = 34; arrows indicate parallel lines (gh || hj? wait, no, the arrows on gk and hj suggest they are parallel, so triangle gki and triangle hji are similar?)

ij = \boxed{}

Explanation:

Step1: Identify Similar Triangles

The lines \( HJ \) and \( GK \) are parallel (indicated by the arrows), so triangles \( IHJ \) and \( IGK \) are similar by the Basic Proportionality Theorem (Thales' theorem).

Step2: Set Up Proportion

Let \( IJ = x \) and \( IK = 34 + x \)? Wait, no, actually \( G I = 63 \), \( G H = 42 \), so \( H I = 63 - 42 = 21 \). Wait, no, \( G I \) is the total length, \( G H = 42 \), so \( H I = 63 - 42 = 21 \). Since \( \triangle IHJ \sim \triangle IGK \), the ratio of corresponding sides is equal. So \( \frac{HI}{GI} = \frac{IJ}{IK} \). \( HI = 63 - 42 = 21 \), \( GI = 63 \), \( IK = 34 \)? Wait, no, \( KJ = 34 \), so \( IK = IJ + JK \)? Wait, maybe I got the sides wrong. Wait, the sides \( GH \) and \( HI \): \( GH = 42 \), \( GI = 63 \), so \( HI = 63 - 42 = 21 \). The sides \( KJ \) and \( IK \): Wait, the triangles are \( \triangle IHJ \) and \( \triangle IGK \), so \( HJ \parallel GK \), so \( \angle IHJ = \angle IGK \) and \( \angle IJ H = \angle IK G \), so similarity by AA. Therefore, the ratio of \( HI \) to \( GI \) is equal to the ratio of \( IJ \) to \( IK \). Wait, \( HI = 21 \), \( GI = 63 \), so the ratio is \( 21/63 = 1/3 \). Then \( IJ / IK = 1/3 \). But \( IK = IJ + JK \), and \( JK = 34 \)? Wait, no, \( JK = 34 \), so \( IK = IJ + 34 \). So \( IJ / (IJ + 34) = 1/3 \). Cross-multiplying: \( 3IJ = IJ + 34 \), \( 2IJ = 34 \), \( IJ = 17 \). Wait, that makes sense. Alternatively, the ratio of \( HI \) to \( GI \) is \( 21/63 = 1/3 \), so the ratio of \( IJ \) to \( IK \) is also \( 1/3 \). But \( IK = KJ + IJ \)? Wait, no, \( KJ \) is 34, so \( IK = IJ + 34 \). So \( IJ / (IJ + 34) = 1/3 \), solving gives \( IJ = 17 \).

Answer:

\( IJ = 17 \)