Question

what is the length of (overline{gd})?
(gd=square)

Explanation:

Step1: Determine the similarity of triangles

Since \(GH\parallel EF\), \(\triangle DGH\sim\triangle DEF\) (by the Basic Proportionality Theorem or AA similarity criterion as corresponding angles are equal).

Step2: Set up the proportion

The ratio of corresponding sides of similar triangles is equal. Let \(GD = x\). Then \(\frac{GD}{ED}=\frac{HD}{FD}\). We know \(ED=EG + GD=6 + x\), \(FD = FH+HD = 8 + 14=22\), and \(HD = 14\). But another way: using the mid - segment (if we assume \(GH\) is a mid - segment, but from the parallel lines and the ratio of sides). Wait, a better approach:
Since \(\triangle DGH\sim\triangle DEF\), \(\frac{GD}{ED}=\frac{HD}{FD}\). But also, if we use the property of parallel lines dividing sides proportionally. Let's use the formula for similar triangles. The ratio of sides: \(\frac{GD}{GD + 6}=\frac{14}{14 + 8}\). Cross - multiply: \((GD+6)\times14=GD\times(14 + 8)\). Expand: \(14GD+84 = 22GD\). Subtract \(14GD\) from both sides: \(84=22GD - 14GD\). So \(8GD = 84\), \(GD=\frac{84}{8}\) is wrong. Wait, no, another approach.
Since \(GH\parallel EF\), \(\frac{GD}{GE}=\frac{HD}{FH}\) (by the converse of the Basic Proportionality Theorem). Substitute \(GE = 6\), \(HD = 14\), \(FH = 8\). Then \(\frac{GD}{6}=\frac{14}{8}\). Cross - multiply: \(8GD=6\times14\).

Step3: Solve for \(GD\)

\(8GD = 84\), so \(GD=\frac{84}{8}=\frac{21}{2}=10.5\)

Answer:

\(10.5\)