Question
If P(x1, y1, z1) and Q(x2 ,y2 ,z2) be the end points of a line segment PQ and let R be a point on PQ dividing PQ in the ratio m : n internally, then prove that the co-ordinates of R are:
Solution
PROOF: Let R(x, y, z) be the points which divides the line segment PQ in the ratio m : n internally. Draw PL, QN and RM perpendiculars to XY-plane as shown in figure. Through R, draw a line SRT parallel to the line LMN, produce LP to S and T lies on the line QN.
Clearly, SLMR and RMNT are parallelograms,
Also are similar.
or
Similarly, we have
Hence, R is
Similarly, for external division in the ratio m : n, the co-ordinate of R:
Also are similar.
or
Similarly, we have
Hence, R is
Similarly, for external division in the ratio m : n, the co-ordinate of R: