The area of the largest rectangle will be 3.
By using properties of similar triangles we can say that
[tex]\frac{EF}{FB} - \frac{AC}{CB}[/tex]
B / (3 - a) = 4/3
3b = 12 - 4a
4a = 12 - 3b
a = (12 - 3b) / 4
Now as we know area of a rectangle is length * width. let us find the area
A = l * w
A = b * ( (2 - 3b) / 4 )
A = 3b - 3[tex]b^{2}[/tex]/4
Now to find maximum value we will differentiate the area and put it equal to zero.
[tex]D^{1}[/tex] = 3 - 6b/4 = 0
6b/4 = 3
6b = 12
b = 2
and
a = (12 - 3b) / 4
a = 6/4
a = 3/2
So, area will be = 3/2 * 2
Therefore maximum are of the triangle will be 3.
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