Difference between revisions of "1988 AJHSME Problems/Problem 9"
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draw((0,b)--(11,b)); | draw((0,b)--(11,b)); | ||
} | } | ||
− | draw((0,6)--(2,6)--(1,4)--cycle,linewidth( | + | draw((0,6)--(2,6)--(1,4)--cycle,linewidth(3)); |
− | draw((3,4)--(3,6)--(5,4)--cycle,linewidth( | + | draw((3,4)--(3,6)--(5,4)--cycle,linewidth(3)); |
− | draw((0,1)--(3,2)--(6,1)--cycle,linewidth( | + | draw((0,1)--(3,2)--(6,1)--cycle,linewidth(3)); |
− | draw((7,4)--(6,6)--(9,4)--cycle,linewidth( | + | draw((7,4)--(6,6)--(9,4)--cycle,linewidth(3)); |
− | draw((8,1)--(9,3)--(10,0)--cycle,linewidth( | + | draw((8,1)--(9,3)--(10,0)--cycle,linewidth(3)); |
</asy> | </asy> | ||
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==Solution== | ==Solution== | ||
− | The first triangle has two [[leg|legs]] of length <math>\sqrt{ | + | 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 21:20, 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?
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.