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Difference between revisions of "1988 AJHSME Problems/Problem 9"

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==Problem==
 
==Problem==
  
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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?
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An [[isosceles triangle]] is a [[triangle]] with two [[edge|sides]] of equal [[length]].  How many of the five triangles on the [[square]] grid below are isosceles?
  
 
<asy>
 
<asy>
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   draw((0,b)--(11,b));
 
   draw((0,b)--(11,b));
 
  }
 
  }
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draw((0,6)--(2,6)--(1,4)--cycle,linewidth(1));
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draw((0,6)--(2,6)--(1,4)--cycle,linewidth(3));
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draw((3,4)--(3,6)--(5,4)--cycle,linewidth(1));
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draw((3,4)--(3,6)--(5,4)--cycle,linewidth(3));
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draw((0,1)--(3,2)--(6,1)--cycle,linewidth(1));
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draw((0,1)--(3,2)--(6,1)--cycle,linewidth(3));
−
draw((7,4)--(6,6)--(9,4)--cycle,linewidth(1));
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draw((7,4)--(6,6)--(9,4)--cycle,linewidth(3));
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draw((8,1)--(9,3)--(10,0)--cycle,linewidth(1));
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draw((8,1)--(9,3)--(10,0)--cycle,linewidth(3));
 
</asy>
 
</asy>
  
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==Solution==
 
==Solution==
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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.
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The first triangle has two [[leg|legs]] of length <math>\sqrt{5}</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{10}</math>, and two legs of the fifth triangle have length <math>\sqrt{5}</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==
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{{AJHSME box|year=1988|num-b=8|num-a=10}}
 
{{AJHSME box|year=1988|num-b=8|num-a=10}}
 
[[Category:Introductory Geometry Problems]]
 
[[Category:Introductory Geometry Problems]]
 +
{{MAA Notice}}

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]

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

Solution

The first triangle has two legs of length $\sqrt{5}$, the second has two legs of length 2, the leg lengths of the third triangle are $2$, $\sqrt{5}$, and $\sqrt{13}$, two legs of the fourth triangle have length $\sqrt{10}$, and two legs of the fifth triangle have length $\sqrt{5}$. Therefore all of the triangles in the diagram except the third are isosceles, and there are $4\Rightarrow \mathrm{(D)}$ 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

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