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Difference between revisions of "1964 AHSME Problems/Problem 36"

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Latest revision as of 18:28, 18 April 2020

Problem

In this figure the radius of the circle is equal to the altitude of the equilateral triangle $ABC$. The circle is made to roll along the side $AB$, remaining tangent to it at a variable point $T$ and intersecting lines $AC$ and $BC$ in variable points $M$ and $N$, respectively. Let $n$ be the number of degrees in arc $MTN$. Then $n$, for all permissible positions of the circle:

$\textbf{(A) }\text{varies from }30^{\circ}\text{ to }90^{\circ}$

$\textbf{(B) }\text{varies from }30^{\circ}\text{ to }60^{\circ}$

$\textbf{(C) }\text{varies from }60^{\circ}\text{ to }90^{\circ}$

$\textbf{(D) }\text{remains constant at }30^{\circ}$

$\textbf{(E) }\text{remains constant at }60^{\circ}$

[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]

Solution

E

See Also

1964 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 35 		Followed by
Problem 37
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All AHSME Problems and Solutions

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