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Question

Find the centroid of a triangle, mid-points of whose sides are (1, 2, -3), (3, 0, 1) and (-1, 1, -4)


Solution

A (x1,y1,z1),B(x2,y2,z2) and C (x3,y3,z3)be the

vertices of the given triangle and let

D(1, 2, -3), E(3, 0, 1) and F(-1, 1, -4) be the mid-point of the sides

BC, CA and AB respectively.

D is the mid point respectively.

D is the mid-point of BC.

x2+x32=1,y2+y32=2,z2+z32=3

x2+x3=2,y2+y3=4,z2+z3 = - 6 (i)

E is the mid-point of CA.

x2+x32=3,y1+y32=0,z1+z32=1

x1+x3=3,y1+y3=0,z1+z3 = 2 (ii)

F is the mid-point of AB.

x1+x22=1,y1+y32=1,z1+z22=4

x1+x2=2,y1+y2=2,z1+z2=-8(iii)

Adding the first three equations we get,

2(x1+x2+x3)=6,2(y1+y2+y3) = 6 and  2(z1+z2+z3)=12

x1+x2+x3=3,y1+y2+y3 = 3 and

  z1+z2+z3=6

 The coordinate of the centroid dis given by

(x1+x2+x33,y1+y2+y33,z1+z2+z33)

(1, 1, -2)


Mathematics
RD Sharma
Standard XI

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