Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1995 AHSME Problems/Problem 21"

(@azjps: Could you move this plz?)
(No difference)

Revision as of 22:24, 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


$\mathrm{(A) \ 1 } \qquad \mathrm{(B) \ 2 } \qquad \mathrm{(C) \ 3 } \qquad \mathrm{(D) \ 4 } \qquad \mathrm{(E) \ 5 }$

Solution

The distance between (4,3) and (-4,-3) is $\sqrt{6^2+8^2}=10$. 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. $\Rightarrow \mathrm{(C)}$

See also

Template:Old AMC12 box

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1995_AHSME_Problems/Problem_21&oldid=22248"