Question

here are triangles abc and def. what is the length of segment df? type your answer in the box. units

Explanation:

Step1: Determine similarity of triangles

Since in \(\triangle ABC\) and \(\triangle DEF\), \(\angle A=\angle D = 68^{\circ}\), \(\angle B=\angle F=44^{\circ}\), then \(\triangle ABC\sim\triangle DFE\) (by AA - Angle - Angle similarity criterion).

Step2: Set up proportion

For similar triangles \(\triangle ABC\) and \(\triangle DFE\), the ratios of corresponding sides are equal. That is \(\frac{AB}{DF}=\frac{AC}{DE}\). We know \(AB = 8\), \(AC = 6\), \(DE=18\). Substituting into the proportion \(\frac{8}{DF}=\frac{6}{18}\).

Step3: Solve for \(DF\)

Cross - multiply: \(6\times DF=8\times18\). Then \(6DF = 144\). Divide both sides by 6: \(DF=\frac{144}{6}=24\).

Answer:

\(24\)