MathDB

In right triangle

\triangle ACE

, we have AC \equal{} 12, CE \equal{} 16, and EA \equal{} 20. Points

B

D

, and

F

are located on

\overline{AC}

\overline{CE}

, and

\overline{EA}

, respectively, so that AB \equal{} 3, CD \equal{} 4, and EF \equal{} 5. What is the ratio of the area of

\triangle DBF

to that of

\triangle ACE

? [asy] size(200);defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=3;
pair C = (0,0); pair E = (16,0); pair A = (0,12); pair F = waypoint(E--A,0.25); pair B = waypoint(A--C,0.25); pair D = waypoint(C--E,0.25);
dot(A);dot(B);dot(C);dot(D);dot(E);dot(F);
label("

A

",A,NW);label("

B

",B,W);label("

C

",C,SW);label("

D

",D,S);label("

E

",E,SE);label("

F

",F,NE);
label("

3

",midpoint(A--B),W); label("

9

",midpoint(B--C),W); label("

4

",midpoint(C--D),S); label("

12

",midpoint(D--E),S); label("

5

",midpoint(E--F),NE); label("

15

",midpoint(F--A),NE);
draw(A--C--E--cycle); draw(B--F--D--cycle);[/asy]

<span class='latex-bold'>(A)</span>\ \frac {1}{4}\qquad <span class='latex-bold'>(B)</span>\ \frac {9}{25}\qquad <span class='latex-bold'>(C)</span>\ \frac {3}{8}\qquad <span class='latex-bold'>(D)</span>\ \frac {11}{25}\qquad <span class='latex-bold'>(E)</span>\ \frac {7}{16}

Problem Statement