AP EAMCET2013MathematicsThree Dimensional Geometry
If D(2,1,0), E(2,0,0) and F(0,1,0) are mid-points of the sides B C, C A and A B of A B C , respectively. Then, the centroid of A B C is
Options
- A( 1 3 , 1 3 , 1 3 )
- B( 4 3 , 2 3 , 0 )
- C(- 1 3 , 1 3 , 1 3 )
- D( 2 3 , 1 3 , 1 3 )
Correct answer
B. ( 4 3 , 2 3 , 0 )
Step-by-step solution
Let A (x₁, y₁, z₁ ), B (x₂, y₂, z₂ ) and C (x₃, y₃, z₃ ) Since, F is the mid-point of A B . . array l x₁+x₂=0 y₁+y₂=2 z₁+z₂=0 array Since, D is the mid-point of B C . . array l x₂+x₃=4 y₂+y₃=2 z₂+z₃=0 array and E is the mid-point of A C . array l x₃+x₁=4 y₃+y₁=0 z₃+z₁=0 array So, . array l x₁+x₂+x₃=4 y₁+y₂+y₃=2 z₁+z₂+z₃=0 array array llll & x₃=4, & y₃=0, & z₃=0 & x₁=0, & y₁=0, & z₁=0 and & x₂=0, & y₂=2, & z₂=0 array Centroid of A B C aligned & = ( x₁+x₂+x₃ 3 , y₁+y₂+y₃ 3 , z₁+z₂+z₃ 3 ) & = ( 4 3 , 2 3 , 0 ) aligned