The largest area of ABPC means the largest area of triangle BPC, that is, the distance from point P to BC line is the largest.

The BC equation is: y=x-3.

Let P(x, y) where 0.

Then the distance from point P to BC line d = | x-y-3 |/ root number 2.

d^2=(x-y-3)^2|/2

Because y=x again? -2x-3

therefore

2d^2=[x(3-x)]^2

Because x & gt00,3-x > 0

X+(3-x) is greater than or equal to 2 times of the radical x(3-x).

So x(3-x) is less than or equal to 9/4.

D 2 is less than or equal to 8 1/32.

D is less than or equal to 9/4 times the root number 2.

Only when x=3-x, that is, x=3/2, is the equal sign taken.

So point P(3/2,-15/4)

The maximum area value of quadrilateral ABPC =4*3/2+ 1/2*3 root number 2*9/4 times root number 2=75/8.