Question

what is the length of ? gd =

Explanation:

Step1: Use the mid - segment theorem

The mid - segment theorem states that the length of the mid - segment of a triangle is half the length of the third side. In triangle \(EFD\), \(GH\) is parallel to \(FD\) and \(EH = HF\), \(EG=GD\). So \(GH\) is the mid - segment of \(\triangle EFD\).

Step2: Calculate the length of \(GD\)

We know that \(EH = HF = 8\), \(ED=EG + GD\) and \(EG = GD\) (because \(GH\) is the mid - segment). Also, from the mid - segment relationship, if we consider another pair of sides. In the figure, assume the relevant triangle properties. Since \(GH\) is related to the sides proportionally. Another way: If we consider the fact that in the given figure (by the property of mid - points in a triangle - like structure where the lines are parallel as indicated by the arrows). We can use the ratio of lengths. Let's assume the basic proportionality (Thales') theorem. But more straightforwardly, if we note that \(GD=\frac{1}{2}\times14\) (by mid - segment property in a triangle - like structure where the line segments are divided proportionally as per the parallel lines and mid - point conditions shown in the figure).

Answer:

\(7\)