In the diagram, the length of $DC$ is twice the length of $BD.$ What is the area of the triangle $ABC?$ [asy]
draw((0,0)--(-3,0)--(0,4)--cycle);
draw((0,0)--(6,0)--(0,4)--cycle);
label("3",(-1.5,0),N);
label("4",(0,2),E);
label("$A$",(0,4),N);
label("$B$",(-3,0),S);
label("$C$",(6,0),S);
label("$D$",(0,0),S);
draw((0,0.4)--(0.4,0.4)--(0.4,0));
[/asy]
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
In the diagram, the length of $DC$ is twice the length of $BD.$ What is the area of the triangle $ABC?$ [asy]
draw((0,0)--(-3,0)--(0,4)--cycle);
draw((0,0)--(6,0)--(0,4)--cycle);
label("3",(-1.5,0),N);
label("4",(0,2),E);
label("$A$",(0,4),N);
label("$B$",(-3,0),S);
label("$C$",(6,0),S);
label("$D$",(0,0),S);
draw((0,0.4)--(0.4,0.4)--(0.4,0));
[/asy]
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
18
In the diagram, the length of $DC$ is twice the length of $BD.$ What is the area of the triangle $ABC?$ [asy]
draw((0,0)--(-3,0)--(0,4)--cycle);
draw((0,0)--(6,0)--(0,4)--cycle);
label("3",(-1.5,0),N);
label("4",(0,2),E);
label("$A$",(0,4),N);
label("$B$",(-3,0),S);
label("$C$",(6,0),S);
label("$D$",(0,0),S);
draw((0,0.4)--(0.4,0.4)--(0.4,0));
[/asy]
train