Question

in the diagram below, \\( \overline { g h } \\) is parallel to \\( \overline { d e } \\). if \\( g h \\) is 12 less than \\( f h \\), \\( f e = 55 \\), and \\( d e = 33 \\), find the length of \\( \overline { f h } \\). figures are not necessarily drawn to scale. state your answer in simplest radical form, if necessary.

Explanation:

Step1: Use the similarity of triangles

Since \( \overline{GH}\parallel\overline{DE}\), then \(\triangle FGH\sim\triangle FDE\) (by the AA similarity criterion, as \(\angle F\) is common and \(\angle FGH=\angle FDE\) and \(\angle FHG = \angle FED\) because of parallel lines).
The ratio of corresponding sides of similar triangles is equal. So, \(\frac{GH}{DE}=\frac{FH}{FE}\).
Let \(FH = x\). Then \(GH=x - 12\), \(DE = 33\), and \(FE=55\).
Substitute these values into the proportion: \(\frac{x - 12}{33}=\frac{x}{55}\).

Step2: Cross - multiply

Cross - multiplying the proportion \(\frac{x - 12}{33}=\frac{x}{55}\) gives \(55(x - 12)=33x\).
Expand the left - hand side: \(55x-660 = 33x\).

Step3: Solve for \(x\)

Subtract \(33x\) from both sides: \(55x-33x-660=33x - 33x\), which simplifies to \(22x-660 = 0\).
Add \(660\) to both sides: \(22x=660\).
Divide both sides by \(22\): \(x=\frac{660}{22}=30\).

Answer:

The length of \(\overline{FH}\) is \(30\).