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Problem of the Week
Problem D and Solution
Triangled

Problem

Consider the following diagram.

Horizontal line segment AE has point B directly below A and point D directly above E. Vertical segments AB and DE and diagonal segment BD, crossing AE at C, form two right-angled triangles with common vertex C.

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

Solution

Since \(\triangle CDE\) is right-angled, we use the Pythagorean theorem to solve for \(CE\). \[CE^2 = CD^2 - DE^2= 17^2 - 8^2= 225\] 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 \(\angle ACB\) and \(\angle DCE\) are opposite angles, then \(\angle ACB=\angle DCE\). We also know that \(\angle CAB=\angle CED=90\degree\), so we can conclude that \(\triangle ABC \sim \triangle EDC\). Then, \[\begin{aligned} \frac{BC}{AC} &= \frac{CD}{CE}\\ \frac{BC}{45} &= \frac{17}{15}\\ BC &= 51 \end{aligned}\] Thus, the length of \(BC\) is \(51\).