Problem of the Week
Problem D and Solution
Triangled

Problem

Consider the following diagram.

If $AE=60$, $DE=8$, and $CD=17$, determine the length of $BC$.

Solution

Since $\mathrm{△}CDE$ is right-angled, we use the Pythagorean theorem to solve for $CE$. $$C{E}^2=C{D}^2-D{E}^2={17}^2-8^2=225$$ Thus $CE=\sqrt{225}=15$, since $CE>0$. The diagram is now updated with all the lengths we know so far.

Since $AC+CE=AE$, it follows that $AC=60-15=45$. Since $\mathrm{\angle }ACB$ and $\mathrm{\angle }DCE$ are opposite angles, then $\mathrm{\angle }ACB=\mathrm{\angle }DCE$. We also know that $\mathrm{\angle }CAB=\mathrm{\angle }CED=90\mathrm{°}$, so we can conclude that $\mathrm{△}ABC\sim \mathrm{△}EDC$. Then, $$\begin{array}{rl}\frac{BC}{AC}& =\frac{CD}{CE}\\ \frac{BC}{45}& =\frac{17}{15}\\ BC& =51\end{array}$$ Thus, the length of $BC$ is $51$.