Let A ( z 1), B ( z 2) and C ( z 3) be the vertices of triangle ABC such that | z 1- z 2|=3, z 1 2= z 2 z 3, z 2 2= z 1 z 3 and z1 z2 z3 ≠ 0. Then which of the following is/are correct?

Question

Let A ( z 1), B ( z 2) and C ( z 3) be the vertices of triangle ABC such that | z 1- z 2|=3, z 1 2= z 2 z 3, z 2 2= z 1 z 3 and z1 z2 z3 ≠ 0. Then which of the following is/are correct? (A) Area of triangle ABC =(9 √3/4)  (B) Circumradius of triangle ABC =√3 (C) Area of triangle ABC =3 (D)  triangle ABC is an isosceles triangle

Tardigrade Admin · Accepted Answer

z 1 2= z 2 z 3 text and z 2 2= z 1 z 3 ⇒ z 3 3= z 1 z 2 ⇒ z 1 2+ z 2 2+ z 3 2= z 1 z 2+ z 2 z 3+ z 3 z 1 ⇒ Δ ABC text is an equilateral triangle.

Q. Let $A (z_{1}), B (z_{2})$ and $C (z_{3})$ be the vertices of $△ A BC$ such that $∣ z_{1} - z_{2} ∣ = 3, z_{1}^{2} = z_{2} z_{3}, z_{2}^{2} = z_{1} z_{3}$ and $z_{1} z_{2} z_{3} \neq = 0$ . Then which of the following is/are correct?

Area of $△ A BC = \frac{9 3}{4}$

Circumradius of $△ A BC = 3$

Area of $△ A BC = 3$

$△ A BC$ is an isosceles triangle

$z_{1}^{2} = z_{2} z_{3} and z_{2}^{2} = z_{1} z_{3}$
$\Rightarrow z_{3}^{3} = z_{1} z_{2}$
$\Rightarrow z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = z_{1} z_{2} + z_{2} z_{3} + z_{3} z_{1}$
$\Rightarrow Δ A BC is an equilateral triangle.$

Q. Let A(z1​),B(z2​) and C(z3​) be the vertices of △ABC such that ∣z1​−z2​∣=3,z1​2=z2​z3​,z2​2=z1​z3​ and z1​z2​z3​=0. Then which of the following is/are correct?

Area of △ABC=493​​

Circumradius of △ABC=3​

Area of △ABC=3

△ABC is an isosceles triangle