Question

Derive an expression for the coordinates of a point that divides the line joining the points A(x, y, z,) & B(x₂ y₂ z₂) internally in the ratio m: n and hence find the mid-point of(1,2,3) & (5,6,7)

Answer 1

Obtain the co-ordinates of the point which divides the line joining A(x₁ y₁, z₁,) and B(x₂, y₂, z₂)

Proof: Let P(x₁, y₁, z₁,) and Q(x₂, y₂, z₂) be the given points. Let R(x, y, z) divide PQ internally in the ratio m:n. Draw PL, QM, RN ⊥r ‘ to XY-plane.” 

∴ PL ∥ RN ∥ QM

∴ PL, QM, RN lie in one plane so that the points L, N, M lie in a straight line which is the intersection of this plane and XY plane, through the point R draw a line AB parallel to the line LM. The line AB will intersect the line LP externally at A and the line MQ at B. 

Triangles APR and BQR are similar