Question

1. what is the length of segment df? (lesson 3 - 9)

a. 3 units
b. \\( \frac { 81 } { 4 } \\) units
c. 36 units
d. 48 units

Explanation:

Step1: Determine similarity of triangles

By AA (Angle - Angle) similarity criterion, since \(\angle A=\angle D = 68^{\circ}\) and \(\angle B=\angle F=44^{\circ}\), \(\triangle ABC\sim\triangle DFE\).

Step2: Set up proportion

For similar triangles \(\triangle ABC\) and \(\triangle DFE\), the ratios of corresponding sides are equal. That is \(\frac{AC}{DF}=\frac{AB}{DE}\).
We know \(AC = 9\), \(AB = 12\), and \(DE=27\). Substituting these values into the proportion \(\frac{9}{DF}=\frac{12}{27}\).

Step3: Solve for \(DF\)

Cross - multiply: \(12\times DF=9\times27\). Then \(DF=\frac{9\times27}{12}=\frac{243}{12}=\frac{81}{4}\) (This is wrong. Let's correct the proportion. The correct proportion is \(\frac{AC}{DF}=\frac{AB}{DE}\) should be \(\frac{AC}{DF}=\frac{AB}{DE}\) (no, actually for \(\triangle ABC\sim\triangle DFE\), \(\frac{AC}{DF}=\frac{AB}{DE}\) is wrong. The correct proportion is \(\frac{AC}{DF}=\frac{AB}{DE}\) (no, \(\triangle ABC\sim\triangle DFE\), so \(\frac{AB}{DE}=\frac{AC}{DF}\). Wait, no: \(\triangle ABC\sim\triangle DFE\), so \(\frac{AB}{DF}=\frac{AC}{DE}\) (no, the order of similarity: \(\triangle ABC\sim\triangle DFE\) means \(A\) corresponds to \(D\), \(B\) corresponds to \(F\), \(C\) corresponds to \(E\). So \(\frac{AB}{DF}=\frac{AC}{DE}\).

Substitute \(AB = 12\), \(AC = 9\), \(DE = 27\) into \(\frac{AB}{DF}=\frac{AC}{DE}\).

\(12\times DE=9\times DF\).

\(DF=\frac{12\times27}{9}\)

\(DF = 36\)

Answer:

C. 36 units