Question

in the diagram below, \\( \overline { e f } \\) is parallel to \\( \overline { b c } \\). if \\( e f \\) is 5 less than \\( e d \\), \\( b d = 55 \\), and \\( b c = 44 \\), find the length of \\( \overline { e d } \\). figures are not necessarily drawn to scale. state your answer in simplest radical form, if necessary.

Explanation:

Step1: Set up variables

Let \( ED = x \). Then \( EF=x - 5 \).
Since \( \overline{EF}\parallel\overline{BC} \), by the AA (Angle - Angle) similarity criterion, \( \triangle DEF\sim\triangle DBC \).
For similar triangles, the ratios of corresponding sides are equal. So, \(\frac{EF}{BC}=\frac{ED}{BD}\).

Step2: Substitute values into the proportion

Substitute \( EF=x - 5 \), \( BC = 44 \), \( ED=x \), and \( BD = 55 \) into \(\frac{EF}{BC}=\frac{ED}{BD}\).
We get \(\frac{x - 5}{44}=\frac{x}{55}\).
Cross - multiply: \(55(x - 5)=44x\).
Expand the left - hand side: \(55x-275 = 44x\).

Step3: Solve for \(x\)

Subtract \(44x\) from both sides: \(55x-44x-275=44x - 44x\).
\(11x-275 = 0\).
Add \(275\) to both sides: \(11x=275\).
Divide both sides by \(11\): \(x=\frac{275}{11}=25\).

Answer:

\(25\)