Let's find the coordinates of A and B.
The interception of line x+3y=6 with x axis (y=0): x+3*0=6, so x=6 and A(6, 0).
The interception of line x+3y=6 with y axis (x=0): 0+3*y=6, so y=2 and B(0, 2).
I. The midpoint of AB is
So we have to find the equation of the median OC through the origin O(0,0) and the point C(3, 1). It has a form y=mx and we have
The equation of median OC is
II. The midpoint of OB is
So we have to find the equation of the median AE through two points A and E. It has a form y=mx+b.
The slope
The y intercept b=1.
The equation of median AE is .
III. The midpoint of OA is
So we have to find the equation of the median BD through two points B and D. It has a form y=mx+b.
The slope
The y intercept b=2.
The equation of median BD is
Answer. (I) (II) (III)
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