Geometry with complex numbers — section formula

It ain’t complex, it’s simple !!

Section Formula:

If P(z) divides the line segment joining A(z_{1}) and B(z_{2}) internally in the ratio m:n, then

z = \frac{mz_{2}+nz_{1}}{m+n}

If the division is external, then z=\frac{mz_{2}-nz_{1}}{m-n}

Proof:

Let z_{1}=x_{1}+iy_{1}, z_{2}=x_{2}+iy_{2}. Then, A \equiv (x_{1},y_{1}) and B \equiv (x_{2},y_{2}).

Let z = x+iy. Then, P \equiv (x,y). We know from co-ordinate geometry,

x = \frac{mx_{2}+nx_{1}}{m+n} and y=\frac{my_{2}+my_{1}}{m+n}

Hence, complex number of P is

z = \frac{mx_{2}+nx_{1}}{m+n}+i\frac{my_{2}+my_{1}}{m+n}

\frac{m(x_{2}+iy_{2})+n(x_{1}+iy_{1})}{m+n}

mz_{2}+nz_{1}

more later,

Nalin Pithwa

 

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