Question

in the diagram below, \\( \overline { f g } \\) is parallel to \\( \overline { c d } \\). if \\( f g \\) is 1 more than \\( e g \\), \\( e d = 6 \\), and \\( c d = 8 \\), find the length of \\( \overline { e g } \\). figures are not necessarily drawn to scale. state your answer in simplest radical form, if necessary.

Explanation:

Step1: Use the similarity of triangles

Since \(FG\parallel CD\), \(\triangle EFG\sim\triangle ECD\) (by the AA similarity criterion, as \(\angle E\) is common and \(\angle EFG=\angle ECD\) and \(\angle EGF = \angle EDC\) because of parallel lines).
For similar triangles, the ratio of corresponding sides is equal. So, \(\frac{FG}{CD}=\frac{EG}{ED}\).
Let \(EG = x\). Then \(FG=x + 1\), \(ED = 6\), and \(CD = 8\).
Substituting into the proportion \(\frac{x + 1}{8}=\frac{x}{6}\).

Step2: Cross - multiply

Cross - multiplying gives \(6(x + 1)=8x\).
Expanding the left - hand side: \(6x+6 = 8x\).

Step3: Solve for \(x\)

Subtract \(6x\) from both sides: \(6=8x - 6x\).
So, \(2x=6\), and \(x = 3\).

Answer:

The length of \(\overline{EG}\) is \(3\).