MathDB

In the given figure hexagon

ABCDEF

is equiangular,

ABJI

and

FEHG

are squares with areas

18

and

32

respectively,

\triangle JBK

is equilateral and

FE=BC

. What is the area of

\triangle KBC

<span class='latex-bold'>(A) </span>6\sqrt{2}\qquad<span class='latex-bold'>(B) </span>9\qquad<span class='latex-bold'>(C) </span>12\qquad<span class='latex-bold'>(D) </span>9\sqrt{2}\qquad<span class='latex-bold'>(E) </span>32

[asy] draw((-4,6*sqrt(2))--(4,6*sqrt(2))); draw((-4,-6*sqrt(2))--(4,-6*sqrt(2))); draw((-8,0)--(-4,6*sqrt(2))); draw((-8,0)--(-4,-6*sqrt(2))); draw((4,6*sqrt(2))--(8,0)); draw((8,0)--(4,-6*sqrt(2))); draw((-4,6*sqrt(2))--(4,6*sqrt(2))--(4,8+6*sqrt(2))--(-4,8+6*sqrt(2))--cycle); draw((-8,0)--(-4,-6*sqrt(2))--(-4-6*sqrt(2),-4-6*sqrt(2))--(-8-6*sqrt(2),-4)--cycle); label("

I

",(-4,8+6*sqrt(2)),dir(100)); label("

J

",(4,8+6*sqrt(2)),dir(80)); label("

A

",(-4,6*sqrt(2)),dir(280)); label("

B

",(4,6*sqrt(2)),dir(250)); label("

C

",(8,0),W); label("

D

",(4,-6*sqrt(2)),NW); label("

E

",(-4,-6*sqrt(2)),NE); label("

F

",(-8,0),E); draw((4,8+6*sqrt(2))--(4,6*sqrt(2))--(4+4*sqrt(3),4+6*sqrt(2))--cycle); label("

K

",(4+4*sqrt(3),4+6*sqrt(2)),E); draw((4+4*sqrt(3),4+6*sqrt(2))--(8,0),dashed); label("

H

",(-4-6*sqrt(2),-4-6*sqrt(2)),S); label("

G

",(-8-6*sqrt(2),-4),W); label("

32

",(-10,-8),N); label("

18

",(0,6*sqrt(2)+2),N); [/asy]

Problem Statement