MathDB

In circle

O

, the midpoint of radius

OX

Q

; at

Q

\overline{AB} \perp \overline{XY}

. The semi-circle with

\overline{AB}

as diameter intersects

\overline{XY}

M

. Line

\overline{AM}

intersects circle

O

C

, and line

\overline{BM}

intersects circle

O

D

. Line

\overline{AD}

is drawn. Then, if the radius of circle

O

r

AD

is: [asy]defaultpen(linewidth(.8pt)); unitsize(2.5cm);
real m = 0; real b = 0;
pair O = origin; pair X = (-1,0); pair Y = (1,0); pair Q = midpoint(O--X); pair A = (Q.x, -1*sqrt(3)/2); pair B = (Q.x, -1*A.y); pair M = (Q.x + sqrt(3)/2,0);
m = (B.y - M.y)/(B.x - M.x); b = (B.y - m*B.x);
pair D = intersectionpoint(Circle(O,1),M--(1.5,1.5*m + b));
m = (A.y - M.y)/(A.x - M.x); b = (A.y - m*A.x);
pair C = intersectionpoint(Circle(O,1),M--(1.5,1.5*m + b));
draw(Circle(O,1)); draw(Arc(Q,sqrt(3)/2,-90,90)); draw(A--B); draw(X--Y); draw(B--D); draw(A--C); draw(A--D); dot(O);dot(M);
label("

B

",B,NW); label("

C

",C,NE); label("

Y

",Y,E); label("

D

",D,SE); label("

A

",A,SW); label("

X

",X,W); label("

Q

",Q,SW); label("

O

",O,SW); label("

M

",M,NE+2N);[/asy]

<span class='latex-bold'>(A)</span>\ r\sqrt {2} \qquad <span class='latex-bold'>(B)</span>\ r\qquad <span class='latex-bold'>(C)</span>\ \text{not a side of an inscribed regular polygon}\qquad <span class='latex-bold'>(D)</span>\ \frac {r\sqrt {3}}{2}\qquad <span class='latex-bold'>(E)</span>\ r\sqrt {3}

Problem Statement