## Introduction to Three-dimensional Geometry

# Section Formula

- The co-ordinates of the point ‘R’ which divides the line segment joining two points P(x
_{1}y_{1}z_{1}) and Q(x_{2}y_{2}z_{2}) 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)

### View the Topic in this video From 13:02 To 22:44

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- The coordinates of any point, which divides the join of points P(x
_{1}, y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2}) 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) - The coordinates of any point, which divides the join of points P(x
_{1}, y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2}) 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) - 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) - Coordinates of the centroid of a triangle formed with vertices P(x
_{1}, y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2}) and R(x_{3}, y_{3}, z_{3}) 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) **Centroid of a Tetrahedron:**If (x_{1}, y_{1}, z_{1}), (x_{2}, y_{2}, z_{2}) (x_{3}, y_{3}, z_{3}) and (x_{4}, y_{4}, z_{4}) 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)