Question

what is the length of segment df?
type your answer in the box.

Explanation:

Step1: Determine similarity of triangles

Since in \(\triangle ABC\), \(\angle A = 68^{\circ}\), \(\angle B=44^{\circ}\), then \(\angle C=180^{\circ}-68^{\circ}-44^{\circ} = 68^{\circ}\). In \(\triangle DEF\), \(\angle D = 68^{\circ}\), \(\angle F=44^{\circ}\), then \(\angle E=180^{\circ}-68^{\circ}-44^{\circ}=68^{\circ}\). By the AA (Angle - Angle) similarity criterion, \(\triangle ABC\sim\triangle DEF\).

Step2: Set up proportion

For similar triangles \(\triangle ABC\) and \(\triangle DEF\), the ratio of corresponding sides is equal. Let \(AB = 8\), \(AC = 6\), \(DE = 18\), and \(DF=x\). The proportion is \(\frac{AB}{DE}=\frac{AC}{DF}\).
Substitute the values: \(\frac{8}{18}=\frac{6}{x}\).
Cross - multiply: \(8x=18\times6\).
\(8x = 108\).
Solve for \(x\): \(x=\frac{108}{8}=\frac{27}{2}=13.5\).

Answer:

\(13.5\)