Question

in the diagram below, cd = 22.5, ce = 28.1, and da = 13.5. find the length of \\(\overline{eb}\\). round your answer to the nearest tenth if necessary.

Explanation:

Step1: Identify Similar Triangles

Since \( DE \parallel AB \), by the Basic Proportionality Theorem (Thales' theorem), \( \triangle CDE \sim \triangle CAB \). So, the ratios of corresponding sides are equal: \( \frac{CD}{CA} = \frac{CE}{CB} \).

First, find \( CA \): \( CA = CD + DA = 22.5 + 13.5 = 36 \).

Let \( EB = x \), then \( CB = CE + EB = 28.1 + x \).

Step2: Set Up Proportion

Substitute into the proportion: \( \frac{22.5}{36} = \frac{28.1}{28.1 + x} \).

Cross - multiply: \( 22.5(28.1 + x)=36\times28.1 \).

Expand left side: \( 22.5\times28.1+22.5x = 36\times28.1 \).

Calculate \( 22.5\times28.1 = 632.25 \) and \( 36\times28.1 = 1011.6 \).

So, \( 632.25+22.5x = 1011.6 \).

Subtract \( 632.25 \) from both sides: \( 22.5x=1011.6 - 632.25=379.35 \).

Step3: Solve for x

Divide both sides by \( 22.5 \): \( x=\frac{379.35}{22.5}=16.86 \approx 16.9 \) (rounded to the nearest tenth).

Answer:

\( 16.9 \)