1998 AHSME Problem 11

Geometry Level 3

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

4 5 2 6

Daniel Hirschberg by Daniel Hirschberg , 39, USA

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2 solutions

Harsh Depal
Mar 23, 2014

O u t o f 4 p o i n t s o f t h e r e c t a n g l e , w e h a v e t o c h o o s e 2 p o i n t s w h i c h w i l l b e t h e e n d p o i n t s o f t h e d i a m e t e r ∴ N u m b e r O f S o l u t i o n s b e c o m e s 4 C 2 T h a t i s e q u a l t o 6 b u t w a i t , w h e n w e c h o o s e t h e c i r c l e w i t h p o i n t s a s e n d p o i n t s o f t h e d i a m e t e r 1 c a s e i s r e p e a t e d b e c a u s e t h e s a m e c i r c l e w i l l p a s s t h r o u g h A l l T h e 4 P o i n t s . S o T h e A n s w e r i s 5 A n d N o t 6 Out\quad of\quad 4\quad points\quad of\quad the\quad rectangle,\\ we\quad have\quad to\quad choose\quad 2\quad points\quad which\quad will\quad \\ be\quad the\quad endpoints\quad of\quad the\quad diameter\\ \therefore \quad Number\quad Of\quad Solutions\quad becomes\quad 4C2\quad \\ That\quad is\quad equal\quad to\quad 6\quad but\quad wait,\\ when\quad we\quad choose\quad the\quad circle\quad with\quad points\quad as\quad \\ endpoints\quad of\quad the\quad diameter\quad 1\quad case\quad is\quad repeated\\ because\quad the\quad same\quad circle\quad will\quad pass\quad through\\ All\quad The\quad 4\quad Points.\\ So\quad The\quad Answer\quad is\quad 5\quad And\quad Not\quad 6

8 Helpful 0 Interesting 0 Brilliant 0 Confused

After reading the comment by Harsh Depal for a few times, I only understand what he means and reason why the answer is five. Let the rectangular be ABCD, I guessed 6 because of 4C2, of course. [AB,AC,AD,BC,BD,CD] But, the circle with diameter AC and BD are same. Hence, only 5 circles. (Both circle with diameter AC and BD have the same radius and origin at the intercept of diagonals of rectangular)

AhGuek Duck - 7 years, 2 months ago

There are 6 pairs of vertices of R, both diagonals determine the same circle, therefore I conclude the answer is 5

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