1964 AHSME Problems/Problem 36: Difference between revisions
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==Problem== | |||
In this figure the radius of the circle is equal to the altitude of the equilateral triangle <math>ABC</math>. The circle is made to roll along the side <math>AB</math>, remaining tangent to it at a variable point <math>T</math> and intersecting lines <math>AC</math> and <math>BC</math> in variable points <math>M</math> and <math>N</math>, respectively. Let <math>n</math> be the number of degrees in arc <math>MTN</math>. Then <math>n</math>, for all permissible positions of the circle: | In this figure the radius of the circle is equal to the altitude of the equilateral triangle <math>ABC</math>. The circle is made to roll along the side <math>AB</math>, remaining tangent to it at a variable point <math>T</math> and intersecting lines <math>AC</math> and <math>BC</math> in variable points <math>M</math> and <math>N</math>, respectively. Let <math>n</math> be the number of degrees in arc <math>MTN</math>. Then <math>n</math>, for all permissible positions of the circle: | ||
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</asy> | </asy> | ||
==See Also== | |||
{{AHSME 40p box|year=1964|num-b=35|num-a=37}} | |||
[[Category:Introductory Algebra Problems]] | |||
{{MAA Notice}} | |||
Revision as of 22:18, 24 July 2019
Problem
In this figure the radius of the circle is equal to the altitude of the equilateral triangle 
. The circle is made to roll along the side 
, remaining tangent to it at a variable point 
and intersecting lines 
and 
in variable points 
and 
, respectively. Let 
be the number of degrees in arc 
. Then , for all permissible positions of the circle:
![[asy] pair A = (0,0), B = (1,0), C = dir(60), T = (2/3,0); pair M = intersectionpoint(A--C,Circle((2/3,sqrt(3)/2),sqrt(3)/2)), N = intersectionpoint(B--C,Circle((2/3,sqrt(3)/2),sqrt(3)/2)); draw((0,0)--(1,0)--dir(60)--cycle); draw(Circle((2/3,sqrt(3)/2),sqrt(3)/2)); label("$A$",A,dir(210)); label("$B$",B,dir(-30)); label("$C$",C,dir(90)); label("$M$",M,dir(190)); label("$N$",N,dir(75)); label("$T$",T,dir(-90)); [/asy]](http://latex.artofproblemsolving.com/c/3/a/c3ab85218dfc3f60dd6499948a1513946dbb2e69.png)
See Also
| 1964 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 35 |
Followed by Problem 37 | |
| 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 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 | ||
| All AHSME Problems and Solutions | ||
These problems are copyrighted © by the Mathematical Association of America. 
