If Az1- Bz2- Cz3 are the vertices of the triangle ABC such that z1 -z2z3 -z2 - 1-2 - i-2 - then triangle ABC is

Given: (z_1 z_2)(z_3 z_2) = ( 1√(2) + nbsp; i√(2) ) ⇒|z_1 z_2z_3 z_2|=1 ⇒|z_1 z_2|=|z_3 z_2| ⇒ AB=BC and (z_1 z_2)(z_3 z_2)=e^iπ/4 ⇒∠ ABC=45^∘ ...

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Byju's Answer

Standard XII

Mathematics

Rotation Concept

If Az1, Bz2, ...

Question

If $A (z_{1}), B (z_{2}), C (z_{3})$ are the vertices of the triangle ABC such that $\frac{(z_{1} - z_{2})}{(z_{3} - z_{2})} = (\frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}})$ , then triangle $A B C$ is

Equilateral

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Right angled

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Isosceles

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Obtuse angled

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Solution

The correct option is C Isosceles
Given $\frac{(z_{1} - z_{2})}{(z_{3} - z_{2})} = (\frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}})$
$\Rightarrow ∣ ∣ ∣ \frac{z_{1} - z_{2}}{z_{3} - z_{2}} ∣ ∣ ∣ = 1 \Rightarrow | z_{1} - z_{2} | = | z_{3} - z_{2} | \Rightarrow A B = B C$
and $\frac{(z_{1} - z_{2})}{(z_{3} - z_{2})} = e^{i π / 4}$
$\Rightarrow ∠ A B C = 45^{\circ}$

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Similar questions

Q. If

A (z_{1}), B (z_{2}), C (z_{3})

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If $A (z_{1}), B (z_{2}), C (z_{3})$ be the vertices of triangle ABC in which $| A B C - ---- -$ = $\frac{π}{4}$ and $\frac{A B}{B C}$ = $\sqrt{2}$ then $z_{2}$ is equal to

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Rotation Concept

Standard XII Mathematics

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If A(z1),B(z2),C(z3) are the vertices of the triangle ABC such that (z1−z2)(z3−z2)=(1√2+i√2), then triangle ABC is

The correct option is C Isosceles Given (z1−z2)(z3−z2)=(1√2+i√2) ⇒∣∣∣z1−z2z3−z2∣∣∣=1⇒|z1−z2|=|z3−z2|⇒AB=BC and (z1−z2)(z3−z2)=eiπ/4 ⇒∠ABC=45∘

If $A (z_{1}), B (z_{2}), C (z_{3})$ are the vertices of the triangle ABC such that $\frac{(z_{1} - z_{2})}{(z_{3} - z_{2})} = (\frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}})$ , then triangle $A B C$ is