Question

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 D, then find (a + b + D).

Answer 1

`(a+b+D)=15`

Explanation:

1) Given three lines `L1 : x+y+2=0, L2 : x-2y+5=0, and L3 : 7x+y-10=0` form a triangle `DeltaABC`, so the intersection points of these three lines are the co-ordinates of the vertices of `DeltaABC`, which can be found by setting the equations of the three lines equal to one another.
1a) let `x+y+2=x-2y+5`
`=> 3y=3, => y=1`
`=> x+y+2=0, => x+1+2=0, => x=-3`
`=>` vertex `color(red)(A=(x_1,y_1)=(-3,1))`
1b) Similarly, let `x+y+2=7x+y-10`
`=> x=2, y=-4`
`=>` vertex `color(red)(B=(x_2,y_2)=(2,-4))`
1c) Similarly, let `x-2y+5=7x+y-10`
`=> x=1, y=3`
`=>` vertex `color(red)(C=(x_3,y_3)=(1,3))`
2) Recall that when the centroid of a triangle is joined to the vertices of a triangle, it divides the triangle into 3 equal areas, and given that areas of `DeltaPAB, DeltaPBC, and DeltaPCA` are equal,
`=> P` is the centroid of `DeltaABC`
`=> (a,b) = ((x_1+x_2+x_3)/3, (y_1+y_2+y_3)/3)`
`=> (color(red)(a,b))=((-3+2+1)/3, (1-4+3)/3)=(color(red)(0,0))`

3) Recall that if `(x_1,y_1), (x_2,y_2) and (x_3,y_3)` are the coordinates of the vertices of a triangle, then the area of the triangle `=1/2[(x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)]`
`=>` Area `DeltaABC=color(red)D=1/2[-3(-4-3)+2(3-1)+1(1+4)]`
`=1/2(21+4+5)=30/2=color(red)15`

Hence `(a+b+D)=0+0+15=15`