Difference between revisions of "1988 AJHSME Problems/Problem 24"
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</asy> | </asy> | ||
− | The square in the first diagram "rolls" clockwise around the fixed regular hexagon until it reaches the bottom. In which position will the solid triangle be in diagram <math>4</math>? | + | The [[square]] in the first diagram "rolls" clockwise around the fixed regular [[hexagon]] until it reaches the bottom. In which position will the solid [[triangle]] be in diagram <math>4</math>? |
<asy> | <asy> | ||
Line 44: | Line 44: | ||
==Solution== | ==Solution== | ||
− | + | ||
+ | |||
+ | === Solution 1 === | ||
+ | |||
+ | Alternately, we can simply keep track of the "bottom" side of the square. In the diagrams below, this bottom side is shown in red. | ||
+ | |||
+ | <asy> | ||
+ | unitsize(15); | ||
+ | for (int a=0; a<6; ++a) | ||
+ | { | ||
+ | draw(2*dir(60a)--2*dir(60a+60),linewidth(1)); | ||
+ | } | ||
+ | draw((1,1.7320508075688772935274463415059)--(1,3.7320508075688772935274463415059)--(-1,3.7320508075688772935274463415059)--(-1,1.7320508075688772935274463415059)--cycle,linewidth(1)); | ||
+ | fill((.4,1.7320508075688772935274463415059)--(0,3.35)--(-.4,1.7320508075688772935274463415059)--cycle,black); | ||
+ | label("1.",(0,-2),S); | ||
+ | draw(arc((1,1.7320508075688772935274463415059),1,90,300,CW)); | ||
+ | draw((1.5,0.86602540378443864676372317075294)--(1.75,1.7)); | ||
+ | draw((1.5,0.86602540378443864676372317075294)--(2.2,1)); | ||
+ | draw((7,0)--(6,1.7320508075688772935274463415059)--(4,1.7320508075688772935274463415059)--(3,0)--(4,-1.7320508075688772935274463415059)--(6,-1.7320508075688772935274463415059)--cycle,linewidth(1)); | ||
+ | draw((7,0)--(6,1.7320508075688772935274463415059)--(7.7320508075688772935274463415059,2.7320508075688772935274463415059)--(8.7320508075688772935274463415059,1)--cycle,linewidth(1)); | ||
+ | label("2.",(5,-2),S); | ||
+ | draw(arc((7,0),1,30,240,CW)); | ||
+ | draw((6.5,-0.86602540378443864676372317075294)--(7.1,-.7)); | ||
+ | draw((6.5,-0.86602540378443864676372317075294)--(6.8,-1.5)); | ||
+ | draw((14,0)--(13,1.7320508075688772935274463415059)--(11,1.7320508075688772935274463415059)--(10,0)--(11,-1.7320508075688772935274463415059)--(13,-1.7320508075688772935274463415059)--cycle,linewidth(1)); | ||
+ | draw((14,0)--(13,-1.7320508075688772935274463415059)--(14.7320508075688772935274463415059,-2.7320508075688772935274463415059)--(15.7320508075688772935274463415059,-1)--cycle,linewidth(1)); | ||
+ | label("3.",(12,-2.5),S); | ||
+ | draw((21,0)--(20,1.7320508075688772935274463415059)--(18,1.7320508075688772935274463415059)--(17,0)--(18,-1.7320508075688772935274463415059)--(20,-1.7320508075688772935274463415059)--cycle,linewidth(1)); | ||
+ | draw((18,-1.7320508075688772935274463415059)--(20,-1.7320508075688772935274463415059)--(20,-3.7320508075688772935274463415059)--(18,-3.7320508075688772935274463415059)--cycle,linewidth(1)); | ||
+ | label("4.",(19,-4),S); | ||
+ | |||
+ | draw((1,1.7320508075688772935274463415059)--(-1,1.7320508075688772935274463415059),linewidth(1)+red); | ||
+ | draw((6,1.7320508075688772935274463415059)--(7.7320508075688772935274463415059,2.7320508075688772935274463415059),linewidth(1)+red); | ||
+ | draw((14.7320508075688772935274463415059,-2.7320508075688772935274463415059)--(15.7320508075688772935274463415059,-1),linewidth(1)+red); | ||
+ | draw((18,-1.7320508075688772935274463415059)--(18,-3.7320508075688772935274463415059),linewidth(1)+red); | ||
+ | |||
+ | </asy> | ||
==See Also== | ==See Also== | ||
− | [[ | + | {{AJHSME box|year=1988|num-b=23|num-a=25}} |
+ | [[Category:Introductory Geometry Problems]] | ||
+ | {{MAA Notice}} |
Latest revision as of 09:05, 11 June 2024
Contents
[hide]Problem
The square in the first diagram "rolls" clockwise around the fixed regular hexagon until it reaches the bottom. In which position will the solid triangle be in diagram ?
Solution
Solution 1
Alternately, we can simply keep track of the "bottom" side of the square. In the diagrams below, this bottom side is shown in red.
See Also
1988 AJHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
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.