Question

If the origin is the centroid of the triangle PQR with vertices P (2 a , 2, 6), Q (–4, 3 b , –10) and R (8, 14, 2 c ), then find the values of a , b and c .

Answer 1

The given vertices of triangle PQR are P=( 2a,2,6 ) , Q=( −4,3b,−10 ) , R=( 8,14,2c ) The centroid C of triangle PQR is at origin. Thus, C=( 0,0,0 ) .

The centroid C( x,y,z ) of a triangle with vertices, ( x 1 , y 1 , z 1 ) , ( x 2 , y 2 , z 2 ) and ( x 3 , y 3 , z 3 ) is given by,

C( x,y,z )=( x 1 + x 2 + x 3 3 , y 1 + y 2 + y 3 3 , z 1 + z 2 + z 3 3 ) (1)

Therefore,

0= ( 2a+8−4 ) 3 0= 2a+4 3 2a+4=0 a= −4 2 a=−2

0= ( 2+14+3b ) 3 0= 16+3b 3 16+3b=0 3b=−16 b= −16 3

0= ( 6+2c−10 ) 3 0= 2c−4 3 2c−4=0 2c=4 c= 4 2 =2

Therefore, the values of a is −2 , b is −16 3 , c is 2 .