Question

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1. in the diagram below, \\( \overline{af} \\), and \\( \overline{db} \\) intersect at c, and \\( \overline{ad} \\) and \\( \overline{fbe} \\) are drawn such that \\( m \angle d = 65^\circ \\), \\( m \angle cbe = 115^\circ \\), \\( dc = 7.2 \\), \\( ac = 9.6 \\), and \\( fc = 21.6 \\). what is the length of \\( \overline{cb} \\)?

Explanation:

Step1: Identify Similar Triangles

First, we check the angles. We know that \( \angle D = 65^\circ \) and \( \angle CBE = 115^\circ \), so \( \angle ABC = 180^\circ - 115^\circ = 65^\circ \) (linear pair). Thus, \( \angle D=\angle ABC = 65^\circ \). Also, \( \angle ACD=\angle FCB \) (vertical angles). So, by AA (Angle - Angle) similarity criterion, \( \triangle ACD \sim \triangle FCB \).

Step2: Set Up Proportion for Similar Triangles

For similar triangles, the ratios of corresponding sides are equal. So, \( \frac{AC}{FC}=\frac{DC}{BC} \). We are given \( AC = 9.6 \), \( FC = 21.6 \), and \( DC = 7.2 \). Let \( BC=x \) (the length we need to find). Substituting the values into the proportion: \( \frac{9.6}{21.6}=\frac{7.2}{x} \).

Step3: Solve for \( x \) (Cross - Multiply)

Cross - multiplying gives us \( 9.6x=21.6\times7.2 \). First, calculate \( 21.6\times7.2 \): \( 21.6\times7.2 = 155.52 \). Then, solve for \( x \): \( x=\frac{155.52}{9.6} \).

Step4: Calculate the Division

\( \frac{155.52}{9.6}=16.2 \).

Answer:

The length of \( \overline{CB} \) is \( 16.2 \).