Question

in the diagram below, ( overline{ef} ) is parallel to ( overline{bc} ). if ( ef ) is 5 less than ( ed ), ( bd = 55 ), and ( bc = 44 ), find the length of ( overline{ed} ). figures are not necessarily drawn to scale. state your answer in simplest radical form, if necessary.

Explanation:

Step1: Use the similarity of triangles

Since \( \overline{EF}\parallel\overline{BC}\), \(\triangle DEF\sim\triangle DBC\) (by AA similarity, as \(\angle DFE=\angle DCB = 90^{\circ}\) and \(\angle D\) is common). Then the ratio of corresponding sides is equal. Let \(ED=x\), then \(EF=x - 5\).
The ratio of sides gives \(\frac{EF}{BC}=\frac{ED}{BD}\).

Step2: Substitute the values

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: \(55x-275 = 44x\).

Step3: Solve for \(x\)

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

Answer:

\(25\)