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Section Formula


Let the two given points be . Let the point divide AC in the given ratio internally. Draw AF, CH and BG perpendicular to the XY-plane. Obviously AF || BG || CH and feet of these perpendiculars lie in a XY-plane. The points F, G and H will lie on a line which is the intersection of the plane containing AF, BG and CH with the XY-plane.

 

Through the point B draw a line DE parallel to the line FH. Line DE will intersect the line FA externally at the point D and the line HC at E, as shown. Also note that quadrilaterals FGBD and GHEB are parallelograms. The triangles ABD and BCE are similar. Therefore,


Similarly, by drawing perpendiculars to the XZ and YZ-planes, we get


Hence, the coordinates of the point B which divides the line segment joining two points internally in the ratio

Corollary

  1. If the point B divides the line AC externally in the ratio then its coordinates are got by replacing Thus the coordinates of the point B in this case are .
  2. If the point B is the midpoint of the line AC, then . Thus the coordinates of the point B in this case are .These are the coordinates of the mid point of the line segment joining .
  3. Suppose B divides AC in the ratio Its coordinates can be got by taking . Thus the coordinates of the point B in this case are .
Example 1
Find the coordinates of the point which divides the line segment joining the points in the ratio 2:3 (i) internally, (ii) externally

Solution:
Let the required point be
  1. the coordinates of the point which divides the line segment internally is given by

the required point is
  1. the coordinates of the point which divides the line segment externally is given by

the required points

Example 2
Find the ratio in which the line segment (2, 4, 5) and (3, 5, 4) is divided by YZ - plane

Solution:
let the YZ - plane divide the line segment joining in the ration then the coordinates of P are given by


Since P lies on the YZ - plane; its - coordinates is zero

YZ - plane divides AB externally in the ratio 2 : 3

Example 3
A point R with -coordinate 4 lies on the line segment joining the points . Find the coordinates of the point R

Solution:
Suppose R divides PQ in the ratio The coordinates of the point R are given by
Given the -coordinate of the point R is 4

coordinate
  coordinates
  coordinate
the coordinates of the point R are




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