If (-2, 3), (4, -3) and (4, 5) are the mid-points of the sides of a triangle,find the coordinates of its centroid.

Let P(-2, 3), Q(4, -3), R(4, 5) be the mid-points of sides AB, BC and CA respectively of a triangle ABC. Let A(x ₁ , y ₁ ), B(x ₂ , y ₂ ) and C(x ₃ , y ₃ ) be the vertices of triangle ABC. Then,

We know that the midpoint of a line segment AB with A(x, y) and B(u, v) is

So, the midpoint of AB:

⇒ x ₁ + x ₂ = -4 ………………………..(i)
and y ₁ + y ₂ = 6………………………..(ii)

The midpoint of BC:

⇒ x ₂ + x ₃ = 8 ………………………..(iii)
and y ₂ + y ₃ = -6 ………………………..(iv)

The midpoint of CA:

⇒ x ₃ + x ₁ = 8 ………………………..(v)
and y ₃ + y ₁ = 10 ………………………..(vi)
Adding the equations (i), (iii), (v) we get,
2(x ₁ + x ₂ + x ₃ ) = - 4 + 8 + 8
⇒ x ₁ + x ₂ + x ₃ = 6

Adding the equations (ii), (iv), (vi) we get,
2(y ₁ + y ₂ + y ₃ ) = 6 - 6 + 10
⇒ y ₁ + y ₂ + y ₃ = 5

Therefore the coordinates of the centroid of Δ ABC are