Question

in the figure below, \\( \overline { b c } \\) is a bisector of \\( \overline { d e } \\), \\( \overline { b c } = 7 \\) feet, and \\( \overline { d e } = 80 x \\) feet. what is the length of \\( \overline { a d } \\), in feet? a ( 10 x ) b ( 20 x ) c ( 30 x ) d ( 40 x )

Explanation:

Step1: Use the property of similar triangles

Since \( \overline{BC}\) is a bisector of \( \overline{DE}\), we can use the Angle - Angle (AA) similarity criterion. \(\angle BAC=\angle DAE\) (vertically opposite angles) and \(\angle B=\angle D\) (alternate interior angles for parallel lines, assume \(BC\parallel ED\) as per the bisector - segment relation). So, \(\triangle ABC\sim\triangle ADE\).
The ratio of corresponding sides of similar triangles is equal. Let \(AD = x\) (in terms of the given \(DE = 80x\) and \(BC = 7\)). The ratio of sides in similar triangles \(\triangle ABC\) and \(\triangle ADE\) gives \(\frac{BC}{DE}=\frac{1}{2}\) (because of the bisector property, the segment \(BC\) divides \(DE\) proportionally).

Step2: Calculate the length of \(AD\)

We know that if \(\triangle ABC\sim\triangle ADE\), then \(AD=\frac{1}{2}DE\) (by the property of similar triangles and the bisector - segment relation). Given \(DE = 80x\), then \(AD=\frac{80x}{4}=20x\) (using the mid - segment or bisector - segment ratio in similar triangles, the ratio of \(AD\) to \(DE\) is \(1:4\) when considering the construction of the bisector and similar triangle side - length ratios).

Answer:

B. \(20x\)