Question

trapezoid abcd is shown below. \\( \overline { a c } \\) and \\( \overline { b d } \\) intersect at point e. in the diagram, \\( \overline { a b } \\) is 6 units long, \\( \overline { b e } \\) is 4 units long, and \\( \overline { d e } \\) is 16 units long. what is the length of \\( \overline { d c } \\)? units

Explanation:

Step1: Identify similar triangles

Since \(AB\parallel DC\) in trapezoid \(ABCD\), \(\triangle AEB\sim\triangle CED\) (by AA similarity criterion as \(\angle AEB=\angle CED\) (vertical angles) and \(\angle EAB = \angle ECD\), \(\angle EBA=\angle EDC\) (alternate - interior angles for parallel lines \(AB\) and \(DC\))).

Step2: Use the property of similar triangles

For similar triangles \(\triangle AEB\) and \(\triangle CED\), the ratios of corresponding sides are equal. That is \(\frac{AB}{DC}=\frac{BE}{DE}\).
We know that \(AB = 6\), \(BE = 4\), and \(DE=16\). Substituting these values into the proportion \(\frac{6}{DC}=\frac{4}{16}\).
Cross - multiply: \(4\times DC=6\times16\).

Step3: Solve for \(DC\)

$$
LATEXBLOCK0
$$

Answer:

\(24\)