1965 AHSME Problems/Problem 25

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

Problem

Let $ABCD$ be a quadrilateral with $AB$ extended to $E$ so that $\overline{AB} = \overline{BE}$. Lines $AC$ and $CE$ are drawn to form $\angle{ACE}$. For this angle to be a right angle it is necessary that quadrilateral $ABCD$ have:

$\textbf{(A)}\ \text{all angles equal} \qquad \textbf{(B) }\ \text{all sides equal} \\ \textbf{(C) }\ \text{two pairs of equal sides} \qquad \textbf{(D) }\ \text{one pair of equal sides} \\ \textbf{(E) }\ \text{one pair of equal angles}$

Solution

$[asy] draw((0,0)--(16,0)); dot((0,0)); label("A", (-1,-1)); dot((8,0)); label("B",(8,-1)); dot((16,0)); label("E",(17,-1)); draw((0,0)--(8*sqrt(3),4)--(16,0)); draw((8,0)--(8*sqrt(3),4)); dot((8*sqrt(3),4)); label("C", (8*sqrt(3)+0.75,5)); draw((0,0)--(2,6)--(8*sqrt(3),4)); dot((2,6)); label("D", (1,7)); markscalefactor=0.1; draw(rightanglemark((0,0), (8*sqrt(3),4), (16,0))); [/asy]$

Because $\triangle ACE$ is right, the midpoint of its hypoteneuse (namely, $B$) is its orthocenter. Thus, $AB=BC$, and so two side lengths of quadrilateral $ABCD$ are equal. The placement of $D$ is irrelevant. Thus, our answer is $\fbox{\textbf{(D) }one pair of equal sides}$.