The coordinates of the mid-points of sides AB, BC and CA of ΔABC are D(1, 2, -3), E(3, 0, 1) and F(-1, 1, -4) respectively. Write the coordinates of its centroid.

Given: The mid-points of the sides of the triangle are P(1, 2, -3), Q(3, 0, 1) and R(-1, 1, -4).


To find: the coordinates of the centroid


Formula used:


Centroid of triangle ABC whose vertices are A(x1, y1, z1), B(x2, y2, z2) and C(x3, y3, z3) is given by,



Section Formula:


A line AB is divided by C in m:n where A(x, y, z) and B(a, b, c).


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The coordinates of C is given by,



We know the mid-point divides side in the ratio of 1:1.


Therefore,


The coordinates of C is given by,



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P(1, 2, -3) is mid-point of A(x1, y1, z1) and B(x2, y2, z2)


Therefore,




Q(3, 0, 1) is mid-point of B(x2, y2, z2) and C(x3, y3, z3)


Therefore,




R(-1, 1, -4) is mid-point of A(x1, y1, z1) and C(x3, y3, z3)


Therefore,




x1 + x2 = 2……………………(1)


x2 + x3 = 6………………………(2)


x1 + x3 = -2……………………(3)


Adding (1), (2) and (3):


⇒ x1 + x2 + x2 + x3 + x1 + x3 = 2 + 6 – 2


⇒ 2x1 + 2x2 + 2x3 = 6


⇒ 2(x1 + x2 + x3) = 6


⇒ x1 + x2 + x3 = 3


y1 + y2 = 4……………………(4)


y2 + y3 = 0……………………(5)


y1 + y3 = 2……………………(6)


Adding (4), (5) and (6):


⇒ y1 + y2 + y2 + y3 + y1 + y3 = 4 + 0 + 2


⇒ 2y1 + 2y2 + 2y3 = 6


⇒ 2(y1 + y2 + y3) = 6


⇒ y1 + y2 + y3 = 3


z1 + z2 = -6……………………(7)


z2 + z3 = 2……………………(8)


z1 + z3 = -8……………………(9)


Adding (7), (8) and (9):


⇒ z1 + z2 + z2 + z3 + z1 + z3 = -6 + 2 – 8


⇒ 2z1 + 2z2 + 2z3 = -12


⇒ 2(z1 + z2 + z3) = -12


⇒ z1 + z2 + z3 = -6


Centroid of the triangle




= (1, 1, -2)


Hence, the centroid of the triangle is (1, 1, -2)


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