If the complex numbers z1, z2, z3, are the vertices of a parallelogram A B C D, then the fourth vertex is

Question

If the complex numbers z1, z2, z3, are the vertices of a parallelogram A B C D, then the fourth vertex is (A) (1/2)(z1+z2) (B) (1/4)(z1+z2-z3-z4) (C) (1/3)(z1+z2+z3) (D) z1+z3-z2

Tardigrade Admin · Accepted Answer

As A B C D is a parallelogram, mid point of A C= mid point of B D ⇒ (1/2)(z1+z3)=(1/2)(z2+z4) ⇒ z4=z1+z3-z2

Q. If the complex numbers $z_{1}, z_{2}, z_{3}$ , are the vertices of a parallelogram $A BC D$ , then the fourth vertex is

$\frac{1}{2} (z_{1} + z_{2})$

$\frac{1}{4} (z_{1} + z_{2} - z_{3} - z_{4})$

$\frac{1}{3} (z_{1} + z_{2} + z_{3})$

$z_{1} + z_{3} - z_{2}$

As $A BC D$ is a parallelogram, mid point of $A C =$ mid point of $B D$
$\Rightarrow \frac{1}{2} (z_{1} + z_{3}) = \frac{1}{2} (z_{2} + z_{4})$
$\Rightarrow z_{4} = z_{1} + z_{3} - z_{2}$

Q. If the complex numbers z1​,z2​,z3​, are the vertices of a parallelogram ABCD, then the fourth vertex is

21​(z1​+z2​)

41​(z1​+z2​−z3​−z4​)

31​(z1​+z2​+z3​)

z1​+z3​−z2​