Question

applying the side - splitter theorem
what is the length of \\( \overline{cd} \\)?
\\( gd= \\)

Explanation:

Step1: Apply the Side - Splitter Theorem

The Side - Splitter Theorem states that if a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. In \(\triangle EFD\), since \(GH\parallel EF\), we have the proportion \(\frac{EG}{GD}=\frac{FH}{HD}\).

Step2: Substitute the known values

We know that \(EG = 6\), \(FH=8\), and \(HD = 14\). Let \(GD=x\). Then the proportion becomes \(\frac{6}{x}=\frac{8}{14}\).

Step3: Cross - multiply to solve for \(x\)

Cross - multiplying gives us \(8x=6\times14\). So, \(8x = 84\).

Step4: Solve for \(x\)

Divide both sides by 8: \(x=\frac{84}{8}=\frac{21}{2}=10.5\).

Answer:

\(10.5\)