Difference between revisions of "2009 AMC 8 Problems/Problem 20"
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| + | ==Problem== | ||
How many non-congruent triangles have vertices at three of the eight points in the array shown below? | How many non-congruent triangles have vertices at three of the eight points in the array shown below? | ||
<asy>dot((0,0)); | <asy>dot((0,0)); | ||
| − | dot(( | + | dot((.5,.5)); |
dot((.5,0)); | dot((.5,0)); | ||
| − | dot((. | + | dot((.0,.5)); |
dot((1,0)); | dot((1,0)); | ||
dot((1,.5)); | dot((1,.5)); | ||
| Line 11: | Line 12: | ||
<math> \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9</math> | <math> \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9</math> | ||
| + | |||
| + | ==Solution== | ||
| + | Assume the base of the triangle is on the bottom four points because a congruent triangle can be made by reflecting the base on the top four points. For a triangle with a base of length <math>1</math>, there are <math>3</math> triangles. For a triangle with a base of length <math>2</math>, there are <math>3</math> triangles. For length <math>3</math>, there are <math>2</math>. In total, the number of non-congruent triangles is <math>3+3+2=\boxed{\textbf{(D)}\ 8}</math>. | ||
| + | |||
| + | ==See Also== | ||
| + | {{AMC8 box|year=2009|num-b=19|num-a=21}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 21:18, 11 February 2020
Problem
How many non-congruent triangles have vertices at three of the eight points in the array shown below?
![[asy]dot((0,0)); dot((.5,.5)); dot((.5,0)); dot((.0,.5)); dot((1,0)); dot((1,.5)); dot((1.5,0)); dot((1.5,.5));[/asy]](http://latex.artofproblemsolving.com/9/d/f/9df9390740014533082572b1aac7c25c4bedfb57.png)
Solution
Assume the base of the triangle is on the bottom four points because a congruent triangle can be made by reflecting the base on the top four points. For a triangle with a base of length 
, there are 
triangles. For a triangle with a base of length 
, there are triangles. For length , there are . In total, the number of non-congruent triangles is 
.
See Also
| 2009 AMC 8 (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 | ||
| All AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
