Question

If z1, z2, z3 are distinct complex numbers such that 3/|z2-z3| = 4/|z3-z1| = 5/|z1-z2|

Answer 1

Answer:

Step-by-step explanation:

Concept:

The distance of the complex numbers $z_{1}\:$and$\:z_{2}$ in argand diagram is

$|z_{1}-z_{2}|$

Given:

$\frac{3}{|z_{2}-z_{3}|}=\frac{4}{|z_{3}-z_{1}|}=\frac{5}{|z_{1}-z_{2}|}=k\:(say)$

Then

$\frac{3}{|z_{2}-z_{3}|}=k,\:\frac{4}{|z_{3}-z_{1}|}=k,\:\frac{5}{|z_{1}-z_{2}|}=k$

$|z_{2}-z_{3}|=\frac{3}{k},\:|z_{3}-z_{1}|=\frac{4}{k},\:|z_{1}-z_{2}|=\frac{5}{k}$

Now,

$|z_{2}-z_{3}|^2+|z_{3}-z_{1}|^2\\\\=\frac{9}{k^2}+\frac{16}{k^2}\\\\=\frac{25}{k^2}\\\\=|z_{1}-z_{2}|^2$

so, sum of the squares of two sides of a triangle is equal to square of the third side.

Hence, the triangle formed by these numbers is a right angled triangle.