1998 AHSME Problems/Problem 11

Problem

Let $R$ be a rectangle. How many circles in the plane of $R$ have a diameter both of whose endpoints are vertices of $R$ ?

$\mathrm{(A) \ }1 \qquad \mathrm{(B) \ }2 \qquad \mathrm{(C) \ }4 \qquad \mathrm{(D) \ }5 \qquad \mathrm{(E) \ }6$

Solution

There are $6$ pairs of vertices of $R$ . However, both diagonals determine the same circle, therefore the answer is $\boxed{5}$ .

$[asy] size(200); defaultpen(0.8); pair A=(0,0), B=(5,0), C=(5,2), D=(0,2); draw ( A--B--C--D--cycle ); draw( circle( (A+B)/2, length((A-B)/2) ), red ); draw( circle( (C+B)/2, length((C-B)/2) ), red ); draw( circle( (C+D)/2, length((C-D)/2) ), red ); draw( circle( (A+D)/2, length((A-D)/2) ), red ); draw( circle( (A+B+C+D)/4, length((A-C)/2) ), green ); [/asy]$

1998 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 10	Followed by Problem 12
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1998 AHSME Problems/Problem 11

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