In triangle ABC the mid point of the sides AB, BC and AC respectively (l, 0, 0), (0, m, 0) and (0, 0, n). Then, `(AB^2+BC^2+CA^2)/(l^2+m^+n^2)` is equal to

Options

A. 2
B. 4
C. 8
D. 16

Answer

C

Explanation

We know that in a `DeltaABC` if the coordinates of the mid-points of sides BC,CA and AB are respectively `D(x_(1),y_(1),z_(1))), E(x_(2),y_(2),z_(2))` and `F(x_(3),y_(3),z_(3))`,then the coordinates of its vertices are `A(-x_(1)+x_(2)+x_(3),-y_(1)+y_(2)+y_(3),-z_(1)+z_(2)+z_(3)),`
`B(x_(1)-x_(2)+x_(3),y_(1)-y_(2)+y_(3),z_(1)-z_(2)+z_(3))` and
`C(x_(1)+x_(2)-x_(3),y_(1)+y_(2)-y_(3),z_(1)+z_(2)-z_(3))`
Thus, the coordinates of A,B,C are `(l,-m,n),B(l,m,-n)` and `C(-l,m,n)`
`:.(AB^(2)+BC^(2)+CA^(2))/(l^(2)+m^(2)+n^(2))=(4(m^(2)+n^(2))+4(l^(2)+n^(2))+4(l^(2)+m^(2)))/(l^(2)+m^(2)+n^(2))=8`