Question

in the diagram below, \\(\overline{de}\\) is parallel to \\(\overline{ab}\\). if \\(dc = 9\\), \\(ad = 7.2\\), and \\(ce = 7\\), find the length of \\(\overline{cb}\\). figures are not necessarily drawn to scale.

Explanation:

Step1: Identify Similar Triangles

Since \( \overline{DE} \parallel \overline{AB} \), by the Basic Proportionality Theorem (Thales' theorem), \( \triangle CDE \sim \triangle CAB \) (AA similarity, as both have a right angle and share \( \angle C \)).

Step2: Set Up Proportion

For similar triangles, the ratios of corresponding sides are equal. So, \( \frac{DC}{AC} = \frac{CE}{CB} \). First, find \( AC \): \( AC = AD + DC = 7.2 + 9 = 16.2 \).

Step3: Solve for \( CB \)

Substitute values into the proportion: \( \frac{9}{16.2} = \frac{7}{CB} \). Cross - multiply: \( 9 \times CB = 16.2 \times 7 \). Then, \( CB=\frac{16.2\times7}{9} \). Calculate \( 16.2\times7 = 113.4 \), and \( \frac{113.4}{9}=12.6 \).

Answer:

\( 12.6 \)