Q. 115.0( 1 Vote )

# Find the centroid

Answer :

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).

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,

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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