Difference between revisions of "1988 AJHSME Problems/Problem 9"
(New page: ==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) { ...) |
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==Problem== | ==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? | + | An [[isosceles triangle]] is a [[triangle]] with two [[Side|sides]] of equal [[length]]. How many of the five triangles on the [[square]] grid below are isosceles? |
<asy> | <asy> | ||
| Line 22: | Line 22: | ||
==Solution== | ==Solution== | ||
| − | The first triangle has two legs of length <math>\sqrt{2^2+1^2}</math>, the second has two legs of length 2, the leg lengths of the third triangle are <math>2</math>, <math>\sqrt{5}</math>, and <math>\sqrt{13}</math>, two legs of the fourth triangle have length <math>\sqrt{3^2+1^2}</math>, and two legs of the fifth triangle have length <math>\sqrt{1^2+2^2}</math>. Therefore all of the triangles in the diagram except the third are isosceles, and there are <math>4\Rightarrow \mathrm{(D)}</math> are isosceles. | + | The first triangle has two [[leg|legs]] of length <math>\sqrt{2^2+1^2}</math>, the second has two legs of length 2, the leg lengths of the third triangle are <math>2</math>, <math>\sqrt{5}</math>, and <math>\sqrt{13}</math>, two legs of the fourth triangle have length <math>\sqrt{3^2+1^2}</math>, and two legs of the fifth triangle have length <math>\sqrt{1^2+2^2}</math>. Therefore all of the triangles in the diagram except the third are isosceles, and there are <math>4\Rightarrow \mathrm{(D)}</math> are isosceles. |
==See Also== | ==See Also== | ||
| − | [[ | + | {{AJHSME box|year=1988|num-b=8|num-a=10}} |
| + | [[Category:Introductory Geometry Problems]] | ||
Revision as of 16:47, 2 June 2009
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(1)); draw((3,4)--(3,6)--(5,4)--cycle,linewidth(1)); draw((0,1)--(3,2)--(6,1)--cycle,linewidth(1)); draw((7,4)--(6,6)--(9,4)--cycle,linewidth(1)); draw((8,1)--(9,3)--(10,0)--cycle,linewidth(1)); [/asy]](http://latex.artofproblemsolving.com/f/3/9/f3988b8c0c8684691734c8ec4dc94ee2e6a375a6.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 | ||
