Question

1. in the diagram, lines bc and de are both vertical.

what is the length of ad?
a. 4.5
b. 5
c. 7.5
d. 10
of the sides of triangle abc.

Explanation:

Step1: Identify Similar Triangles

Since \( BC \) and \( DE \) are vertical, \( BC \parallel DE \). Thus, \( \triangle ABC \sim \triangle ADE \) by the AA (Angle - Angle) similarity criterion (both triangles share \( \angle A \) and have right angles at \( B \) and \( D \) respectively).

Step2: Set Up Proportion

For similar triangles, the ratios of corresponding sides are equal. So, \( \frac{AB}{AD}=\frac{BC}{DE} \). Let \( AD = x \). We know \( AB = 3 \), \( BC = 2 \), and \( DE = 5 \). Substituting these values, we get \( \frac{3}{x}=\frac{2}{5} \).

Step3: Solve for \( x \)

Cross - multiply the proportion \( \frac{3}{x}=\frac{2}{5} \). We have \( 2x=3\times5 \), which simplifies to \( 2x = 15 \). Then, divide both sides by 2: \( x=\frac{15}{2}=7.5 \).

Answer:

C. 7.5