Question

in the figure above, \\( \overline { b d } \\) is parallel to \\( \overline { a e } \\). what is the length of \\( \overline { c e } \\)?

Explanation:

Step1: Use the similarity of triangles

Since \(\overline{BD}\parallel\overline{AE}\), \(\triangle CBD\sim\triangle CAE\).
The ratio of corresponding sides of similar triangles is equal. Let \(CE = x\). Then \(\frac{BD}{AE}=\frac{CB}{CA}\).
We know \(BD = 6\), \(AE=18\), \(CB = 8\), and \(CA=CB + BA\). Also, from the ratio \(\frac{BD}{AE}=\frac{CB}{CE}\) (because of similar - triangle property \(\frac{a}{b}=\frac{c}{d}\) for \(\triangle CBD\) and \(\triangle CAE\)).

Step2: Set up the proportion

Substitute the values into the proportion \(\frac{6}{18}=\frac{8}{x}\).
Cross - multiply: \(6x=18\times8\).

Step3: Solve for \(x\)

\(6x = 144\), then \(x=\frac{144}{6}\).

Answer:

\(24\)