Question

question
in the diagram below, \\(\overline{jk}\\) is parallel to \\(\overline{gh}\\). if \\(ik = 6\\), \\(hk = 3\\), and \\(ji = 8\\), find the length of \\(\overline{gi}\\). figures are not necessarily drawn to scale.

Explanation:

Step1: Identify Similar Triangles

Since \( \overline{JK} \parallel \overline{GH} \), by the Basic Proportionality Theorem (Thales' theorem), \( \triangle IJK \sim \triangle IGH \). So, the ratios of corresponding sides are equal: \( \frac{IJ}{IG} = \frac{IK}{IH} \).

Step2: Calculate \( IH \)

\( IH = IK + HK = 6 + 3 = 9 \). Let \( GI = x \), then \( IG = IJ + JG = 8 + (x - 8) \)? Wait, no—wait, \( IJ = 8 \), let \( GI = x \), so \( IG = IJ + JG \)? Wait, no, actually, \( IJ \) is a segment on \( IG \)? Wait, looking at the diagram, \( I \) to \( J \) to \( G \), so \( IG = IJ + JG \), but \( IJ = 8 \), let \( JG = y \), so \( IG = 8 + y \). Wait, no, the similarity: \( \triangle IJK \sim \triangle IGH \), so \( \frac{IJ}{IG} = \frac{IK}{IH} \). So \( IJ = 8 \), \( IK = 6 \), \( IH = 6 + 3 = 9 \). Let \( IG = x \), then \( \frac{8}{x} = \frac{6}{9} \).

Step3: Solve for \( x \)

Cross - multiply: \( 6x = 8\times9 \). \( 6x = 72 \). Divide both sides by 6: \( x=\frac{72}{6}=16 \).

Answer:

The length of \( \overline{GI} \) is \( 16 \).