Question

opal has already proved that \\( \triangle a d f \sim \triangle a b c \\). they are now attempting to prove that \\( d f=\frac{1}{2} b c \\) and \\( \overline{d f} \\| \overline{b c} \\). complete their reasoning. which option should fill in the blank?
proof because corresponding sides of similar triangles are proportional.__________ multiply both sides of the equation by segment \\( b c \\) to show that \\( d f=\frac{1}{2} b c \\). because corresponding angles of similar triangles are congruent. \\( \angle a d f=\angle a b c \\). if corresponding angles are congruent, then lines are parallel, so \\( \overline{d f} \\| \overline{b c} \\).
\\( \frac{a d}{a b}=\frac{1}{2} \\)
\\( a d=d b \\)
\\( \frac{a f}{a c}=\frac{1}{2} \\)
\\( d f=b c \\)

Explanation:

Step1: Analyze similar - triangle proportion

Since \(\triangle ADF\sim\triangle ABC\), the ratio of corresponding sides is \(\frac{AD}{AB}=\frac{DF}{BC}\).

Step2: Determine the value of \(\frac{AD}{AB}\)

From the figure, \(AD = DB\), so \(AB=AD + DB=2AD\), then \(\frac{AD}{AB}=\frac{AD}{2AD}=\frac{1}{2}\).

Answer:

\(\frac{AD}{AB}=\frac{1}{2}\)