MathDB
Pentagon and circle

Source:

November 21, 2005

Problem Statement

Given regular pentagon ABCDEABCDE, a circle can be drawn that is tangent to DC\overline{DC} at DD and to AB\overline{AB} at AA. The number of degrees in minor arc ADAD is
<spanclass=latexbold>(A)</span> 72<spanclass=latexbold>(B)</span> 108<spanclass=latexbold>(C)</span> 120<spanclass=latexbold>(D)</span> 135<spanclass=latexbold>(E)</span> 144<span class='latex-bold'>(A)</span>\ 72 \qquad <span class='latex-bold'>(B)</span>\ 108 \qquad <span class='latex-bold'>(C)</span>\ 120 \qquad <span class='latex-bold'>(D)</span>\ 135 \qquad <span class='latex-bold'>(E)</span>\ 144
[asy] size(100); defaultpen(linewidth(0.7)); draw(rotate(18)*polygon(5)); real x=0.6180339887; draw(Circle((-x,0), 1)); int i; for(i=0; i<5; i=i+1) { dot(origin+1*dir(36+72*i)); } label("BB", origin+1*dir(36+72*0), dir(origin--origin+1*dir(36+72*0))); label("AA", origin+1*dir(36+72*1), dir(origin--origin+1*dir(36+72))); label("EE", origin+1*dir(36+72*2), dir(origin--origin+1*dir(36+144))); label("DD", origin+1*dir(36+72*3), dir(origin--origin+1*dir(36+72*3))); label("CC", origin+1*dir(36+72*4), dir(origin--origin+1*dir(36+72*4)));[/asy]