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: Apply Basic Proportionality Theorem

Since \(\overline{JK} \parallel \overline{GH}\), by the Basic Proportionality Theorem (Thales' theorem), we have \(\frac{IK}{HK}=\frac{JI}{JG}\).
We know \(IK = 6\), \(HK = 3\), and \(JI = 8\). Substituting these values into the proportion: \(\frac{6}{3}=\frac{8}{JG}\).

Step2: Solve for \(JG\)

Simplify \(\frac{6}{3} = 2\), so the equation becomes \(2=\frac{8}{JG}\). Cross - multiplying gives \(2\times JG=8\), then \(JG = \frac{8}{2}=4\).

Step3: Find the length of \(GI\)

We know that \(GI=JI + JG\). We are given \(JI = 8\) and we found \(JG = 4\). So \(GI=8 + 4=12\).

Answer:

The length of \(\overline{GI}\) is \(\boldsymbol{12}\).