1988 AJHSME Problems/Problem 9

Problem

An isosceles triangle is a triangle with two sides of equal length. How many of the five triangles on the square grid below are isosceles?

for(int a=0; a<12; ++a)
 {
  draw((a,0)--(a,6));
 }
for(int b=0; b<7; ++b)
 {
  draw((0,b)--(11,b));
 }
draw((0,6)--(2,6)--(1,4)--cycle,linewidth(3);
draw((3,4)--(3,6)--(5,4)--cycle,linewidth(3));
draw((0,1)--(3,2)--(6,1)--cycle,linewidth(3));
draw((7,4)--(6,6)--(9,4)--cycle,linewidth(3));
draw((8,1)--(9,3)--(10,0)--cycle,linewidth(3));
 (Error making remote request. Unknown error_msg)

Solution

The first triangle has two legs of length , the second has two legs of length 2, the leg lengths of the third triangle are , , and , two legs of the fourth triangle have length , and two legs of the fifth triangle have length . Therefore all of the triangles in the diagram except the third are isosceles, and there are are isosceles.

See Also

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.