 # Section Formula

• The co-ordinates of the point ‘R’ which divides the line segment joining two points P(x1 y1 z1) and Q(x2 y2 z2) internally and externally in the ratio m : n are given by \tt \left(\frac{mx_{2}+nx_{1}}{m+n},\frac{my_{2}+ny_{1}}{m+n},\frac{mz_{2}+nz_{1}}{m+n}\right) and \tt \left(\frac{mx_{2}-nx_{1}}{m-n},\frac{my_{2}-ny_{1}}{m-n},\frac{mz_{2}-nz_{1}}{m-n}\right)

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1. The coordinates of any point, which divides the join of points P(x1, y1, z1) and Q(x2, y2, z2) in the ratio m : n internally are
\left(\frac{mx_{2}+nx_{1}}{m+n},\frac{my_{2}+ny_{1}}{m+n},\frac{mz_{2}+nz_{1}}{m+n}\right)
2. The coordinates of any point, which divides the join of points P(x1, y1, z1) and Q(x2, y2, z2) in the ratio m : n externally are
\left(\frac{mx_{2}-nx_{1}}{m-n},\frac{my_{2}-ny_{1}}{m-n},\frac{mz_{2}-mz_{1}}{m-n}\right)
3. The coordinates of mid-point of P and Q are
\left(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2},\frac{z_{1}+z_{2}}{2}\right)
4. Coordinates of the centroid of a triangle formed with vertices P(x1, y1, z1) and Q(x2, y2, z2) and R(x3, y3, z3) are
\left(\frac{x_{1}+x_{2}+x_{3}}{3},\frac{y_{1}+y_{2}+y_{3}}{3},\frac{z_{1}+z_{2}+z_{3}}{3}\right)
5. Centroid of a Tetrahedron: If (x1, y1, z1), (x2, y2, z2) (x3, y3, z3) and (x4, y4, z4) are the vertices of a tetrahedron, then its centroid G is given by
\left(\frac{x_{1}+x_{2}+x_{3}+x_{4}}{4},\frac{y_{1}+y_{2}+y_{3}+y_{4}}{4},\frac{z_{1}+z_{2}+z_{3}+z_{4}}{4}\right)