question_answer

ABC is a triangle and AD is the median. If the coordinates of A are (4, 7, -8) and the coordinates of centroid of the triangle ABC are (1, 1, 1), what are the coordinates of D?

A) \[\left( -\frac{1}{2},\,\,2,\,\,11 \right)\]

B) \[\left( -\frac{1}{2},-2,\frac{11}{2} \right)\]

C) \[(-1,2,11)\]

D) \[(-5,-11,19)\]

Correct Answer: B

Solution :

[b] Let coordinated of D be (x, y, z)

Co-ordinated of centroid is (1, 1, 1), and of A, is (4, 7, 8) centroid divides median is 2:1 ratio

\[so,\frac{AO}{OD}=2:1\]

For \[x:1=\frac{2\times x+1\times 4}{1+2}\]

\[\Rightarrow x=-1/2\]

For \[y:1=\frac{2y+1\times 7}{1+2}\]

\[\Rightarrow y=-2\]

For \[z:1=\frac{2\times z+1\times -8}{3}\Rightarrow z=+11/2\]

\[\therefore \]      Co-ordinates of D are (-1/2, -2, 11/2)