Question

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

Explanation:

Step1: Identify similar triangles

$\triangle CDE \sim \triangle CAB$ (AA similarity, shared $\angle C$, $\angle CED = \angle CBA$)

Step2: Set up proportion of sides

$\frac{CD}{CA} = \frac{CE}{CB}$
First, calculate $CA = CD + DA = 22.5 + 13.5 = 36$. Let $EB = x$, so $CB = CE + EB = 28.1 + x$.
Substitute values: $\frac{22.5}{36} = \frac{28.1}{28.1 + x}$

Step3: Solve for $x$

Cross-multiply: $22.5(28.1 + x) = 36 \times 28.1$
Calculate right-hand side: $36 \times 28.1 = 1011.6$
Left-hand side expansion: $22.5 \times 28.1 + 22.5x = 632.25 + 22.5x$
Rearrange: $22.5x = 1011.6 - 632.25 = 379.35$
Solve for $x$: $x = \frac{379.35}{22.5}$

Step4: Compute final value

$x = 16.86 \approx 16.9$

Answer:

16.9