MathDB

In the diagram below, angle

ABC

is a right angle. Point

D

is on

\overline{BC}

, and

\overline{AD}

bisects angle

CAB

. Points

E

and

F

are on

\overline{AB}

and

\overline{AC}

, respectively, so that

AE=3

and

AF=10.

Given that

EB=9

and

FC=27

, find the integer closest to the area of quadrilateral

DCFG.

[asy] size(250); pair A=(0,12), E=(0,8), B=origin, C=(24*sqrt(2),0), D=(6*sqrt(2),0), F=A+10*dir(A--C), G=intersectionpoint(E--F, A--D); draw(A--B--C--A--D^^E--F); pair point=G+1*dir(250); label("

A

", A, dir(point--A)); label("

B

", B, dir(point--B)); label("

C

", C, dir(point--C)); label("

D

", D, dir(point--D)); label("

E

", E, dir(point--E)); label("

F

", F, dir(point--F)); label("

G

", G, dir(point--G)); markscalefactor=0.1; draw(rightanglemark(A,B,C)); label("10", A--F, dir(90)*dir(A--F)); label("27", F--C, dir(90)*dir(F--C)); label("3", (0,10), W); label("9", (0,4), W);[/asy]

Problem Statement