Question

6 practice 6 (from unit 3, lesson 9)
here are triangles abc and def.
what is the length of segment df?
type your answer in the box.
units
how did i do?

Explanation:

Step1: Determine triangle similarity

In \(\triangle ABC\), \(\angle A = 68^{\circ}\), \(\angle B=28^{\circ}\), so \(\angle C=180^{\circ}-68^{\circ}-28^{\circ} = 84^{\circ}\). In \(\triangle DEF\), \(\angle D = 68^{\circ}\), \(\angle F = 28^{\circ}\), so \(\angle E=180^{\circ}-68^{\circ}-28^{\circ}=84^{\circ}\). Since \(\angle A=\angle D\), \(\angle B=\angle F\), \(\angle C=\angle E\), \(\triangle ABC\sim\triangle DFE\) (by AAA 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 \(AB = 4\), \(AC = 2\), \(DE = 8\). Substituting into the proportion \(\frac{4}{DF}=\frac{2}{8}\).

Step3: Solve for \(DF\)

Cross - multiply: \(2\times DF=4\times8\). Then \(2DF = 32\). Divide both sides by 2: \(DF=\frac{32}{2}=16\).

Answer:

\(16\)