Difference between revisions of "1988 AJHSME Problems/Problem 9"
Raymondzhu (talk | contribs) (→Problem) |
Raymondzhu (talk | contribs) (→Problem) |
||
| Line 12: | Line 12: | ||
draw((0,b)--(11,b)); | draw((0,b)--(11,b)); | ||
} | } | ||
| − | draw((0,6)--(2,6)--(1,4)--cycle,linewidth(3); | + | draw((0,6)--(2,6)--(1,4)--cycle,linewidth(3)); |
draw((3,4)--(3,6)--(5,4)--cycle,linewidth(3)); | draw((3,4)--(3,6)--(5,4)--cycle,linewidth(3)); | ||
draw((0,1)--(3,2)--(6,1)--cycle,linewidth(3)); | draw((0,1)--(3,2)--(6,1)--cycle,linewidth(3)); | ||
Latest revision as of 20:20, 15 December 2020
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?
![[asy] 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)); [/asy]](http://latex.artofproblemsolving.com/f/d/a/fdaa909c5f1850e47af241607c3a054eb448b449.png)
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
| 1988 AJHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 8 |
Followed by Problem 10 | |
| 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 | ||
| All AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
