Difference between revisions of "1995 AHSME Problems/Problem 21"
(New page: ==Problem== Two nonadjacent vertices of a rectangle are (4,3) and (-4,-3), and the coordinates of the other two vertices are integers. The number of such rectangles is <math> \mathrm{(A)...) |
(@azjps: Could you move this plz?) |
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==See also== | ==See also== | ||
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Revision as of 21:20, 9 January 2008
Problem
Two nonadjacent vertices of a rectangle are (4,3) and (-4,-3), and the coordinates of the other two vertices are integers. The number of such rectangles is
Solution
The distance between (4,3) and (-4,-3) is 
. Therefore, if you circumscribe a circle around the rectangle, it has a center of (0,0) with a radius of 10/2=5. There are three cases:
Case 1: The point "above" the given diagonal is (4,-3).
Then the point "below" the given diagonal is (-4,3).
Case 2: The point "above" the given diagonal is (0,5).
Then the point "below" the given diagonal is (0,-5).
Case 3: The point "above" the given diagonal is (-5,0).
Then the point "below" the given diagonal is (5,0).
We have only three cases since there are 8 lattice points on the circle. 
See also
| 1995 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 19 |
Followed by Problem 21 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
