MathDB

Let

\overline{AB}

be a diameter of a circle and

C

be a point on

\overline{AB}

with 2 \cdot AC \equal{} BC. Let

D

and

E

be points on the circle such that

\overline{DC} \perp \overline{AB}

and

\overline{DE}

is a second diameter. What is the ratio of the area of

\triangle DCE

to the area of

\triangle ABD

? [asy]unitsize(2.5cm); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=3;
pair O=(0,0), C=(-1/3.0), B=(1,0), A=(-1,0); pair D=dir(aCos(C.x)), E=(-D.x,-D.y);
draw(A--B--D--cycle); draw(D--E--C); draw(unitcircle,white); drawline(D,C); dot(O);
clip(unitcircle); draw(unitcircle); label("

E

",E,SSE); label("

B

",B,E); label("

A

",A,W); label("

D

",D,NNW); label("

C

",C,SW);
draw(rightanglemark(D,C,B,2));[/asy]

<span class='latex-bold'>(A)</span> \ \frac {1}{6} \qquad <span class='latex-bold'>(B)</span> \ \frac {1}{4} \qquad <span class='latex-bold'>(C)</span>\ \frac {1}{3} \qquad <span class='latex-bold'>(D)</span>\ \frac {1}{2} \qquad <span class='latex-bold'>(E)</span>\ \frac {2}{3}

23

Problems(2)