MathDB
Circle Tangent to Special Triangle

Source:

January 18, 2009
analytic geometrygeometry

Problem Statement

A circle with center O O is tangent to the coordinate axes and to the hypotenuse of the 30 30^\circ-60 60^\circ-90 90^\circ triangle ABC ABC as shown, where AB \equal{} 1. To the nearest hundredth, what is the radius of the circle? [asy]defaultpen(linewidth(.8pt)); dotfactor=3;
pair A = origin; pair B = (1,0); pair C = (0,sqrt(3)); pair O = (2.33,2.33);
dot(A);dot(B);dot(C);dot(O);
label("AA",A,SW);label("BB",B,SE);label("CC",C,W);label("OO",O,NW); label("11",midpoint(A--B),S);label("6060^\circ",B,2W + N);
draw((3,0)--A--(0,3)); draw(B--C); draw(Arc(O,2.33,163,288.5));[/asy]<spanclass=latexbold>(A)</span> 2.18<spanclass=latexbold>(B)</span> 2.24<spanclass=latexbold>(C)</span> 2.31<spanclass=latexbold>(D)</span> 2.37<spanclass=latexbold>(E)</span> 2.41 <span class='latex-bold'>(A)</span>\ 2.18\qquad <span class='latex-bold'>(B)</span>\ 2.24\qquad <span class='latex-bold'>(C)</span>\ 2.31\qquad <span class='latex-bold'>(D)</span>\ 2.37\qquad <span class='latex-bold'>(E)</span>\ 2.41