MathDB

36

Part of 1964 AMC 12/AHSME

Problems(1)

Permissible positions of a circle

Source: 1964 AHSME Problem 36

3/19/2014
In this figure the radius of the circle is equal to the altitude of the equilateral triangle ABCABC. The circle is made to roll along the side ABAB, remaining tangent to it at a variable point TT and intersecting lines ACAC and BCBC in variable points MM and NN, respectively. Let nn be the number of degrees in arc MTNMTN. Then nn, for all permissible positions of the circle:
<spanclass=′latex−bold′>(A)</span>varies from 30∘ to 90∘<span class='latex-bold'>(A) </span>\text{varies from }30^{\circ}\text{ to }90^{\circ}
<spanclass=′latex−bold′>(B)</span>varies from 30∘ to 60∘<span class='latex-bold'>(B) </span>\text{varies from }30^{\circ}\text{ to }60^{\circ}
<spanclass=′latex−bold′>(C)</span>varies from 60∘ to 90∘<span class='latex-bold'>(C) </span>\text{varies from }60^{\circ}\text{ to }90^{\circ}
<spanclass=′latex−bold′>(D)</span>remains constant at 30∘<span class='latex-bold'>(D) </span>\text{remains constant at }30^{\circ}
<spanclass=′latex−bold′>(E)</span>remains constant at 60∘<span class='latex-bold'>(E) </span>\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("AA",A,dir(210)); label("BB",B,dir(-30)); label("CC",C,dir(90)); label("MM",M,dir(190)); label("NN",N,dir(75)); label("TT",T,dir(-90)); //Credit to bobthesmartypants for the diagram [/asy]
probabilityAMC