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# A triangle ABC is formed by three lines x+y+2=0,x-2y+5=0 and 7x+y-10=0. P is a point inside the triangle ABC such that areas of the triangles PAB,PBC and PCA are equal. If the co-ordinates of the point P are (a,b) and the area of the triangle ABC is ‘g’​, then find (a+b+g). [P.S. The answer to this question is supposed to be 15.Just tell me how.]

Harsh Hardik
26 Points
4 years ago
To take out the coordinates of A,B and C use the point of intersection of lines
so, coordinates of A come to be x+y=-2 and x-2y=-5 then by using elimination method the coordinates of A come to be (-3,1).
Similarly of B (2,-4) and C (1,3).
Area of triangle ABC
g=1/2[-3(-4-3)+2(3-1)+1(1+4)]
and g comes out to be 15 by using area of triangle.
Now,P is a centroid of the triangle ABC as it divides the triangle into 3 equal areas.
so P(a,b) here x=a and y=b coordinate.
Now, a=x1+x2+x3/3
a=-3+2+1/3=0
similarly of b = 1-4+3/3 = 0
so, a+b+g=0+0+15=15
hope it is clear!!!
Regards Harsh Hardik
Student of fiitjee