# What is the area of the triangle with vertices $A\left( z \right)$, $B\left( {iz} \right)$, and $C\left( {z + iz} \right)$?

A. $\dfrac{1}{2}{\left| {z + iz} \right|^2}$

B. 1

C. $\dfrac{1}{2}$

D. $\dfrac{1}{2}{\left| z \right|^2}$

Answer

Verified

91.2k+ views

**Hint:**The angle between vertex $A\left( z \right)$ and $B\left( {iz} \right)$ is ${90^ \circ }$. So, the base of the triangle is $\left| z \right|$ and the height of the triangle is $\left| {iz} \right|$. By using the area formula of a triangle, we can calculate the area of the $\Delta ABC$.

**Formula Used**:

The angle between two complex numbers ${z_1} = {x_1} + i{y_1}$ and ${z_2} = {x_2} + i{y_2}$ is $\theta = {\tan ^{ - 1}}\left( {\dfrac{{\dfrac{{{y_1}}}{{{x_1}}} - \dfrac{{{y_2}}}{{{x_2}}}}}{{1 + \dfrac{{{y_1}}}{{{x_1}}} \cdot \dfrac{{{y_2}}}{{{x_2}}}}}} \right)$.

The modulus of complex number $z = x + iy$ is $\left| z \right| = \sqrt {{x^2} + {y^2}} $

Area of a triangle$ = \dfrac{1}{2} \times {\rm{base}} \times {\rm{height}}$

**Complete step by step solution:**

Given that the vertices of the triangle are $A\left( z \right)$, $B\left( {iz} \right)$, and $C\left( {z + iz} \right)$.

Image: Triangle ABC

Now putting $z = x + iy$ in vertices

The vertices are $A\left( {x + iy} \right)$, $B\left( {i\left[ {x + iy} \right]} \right)$, and $C\left( {x + iy + i\left[ {x + iy} \right]} \right)$ or $A\left( {x + iy} \right)$, $B\left( { - y + ix} \right)$, and $C\left( {x - y + iy + ix} \right)$. $\because i^{2}=-1$

Now applying the formula $\theta = {\tan ^{ - 1}}\left( {\dfrac{{\dfrac{{{y_1}}}{{{x_1}}} - \dfrac{{{y_2}}}{{{x_2}}}}}{{1 + \dfrac{{{y_1}}}{{{x_1}}} \cdot \dfrac{{{y_2}}}{{{x_2}}}}}} \right)$ to calculate the angle between $A\left( {x + iy} \right)$ and $B\left( { - y + ix} \right)$.

The angle between $OA$ and $OB$ is

$\theta = {\tan ^{ - 1}}\left( {\dfrac{{\dfrac{y}{x} - \left( { - \dfrac{x}{y}} \right)}}{{1 + \dfrac{y}{x} \cdot \left( { - \dfrac{x}{y}} \right)}}} \right)$

$ \Rightarrow \theta= {\tan ^{ - 1}}\left( {\dfrac{{\dfrac{y}{x} - \left( { - \dfrac{x}{y}} \right)}}{{1 - 1}}} \right)$

$ \Rightarrow \theta= {90^ \circ }$ [ Since $\tan {90^ \circ }$ is undefined]

So, the base and height of the triangle are $\left| {OA} \right|$ and $\left| {OB} \right|$.

Apply the modulus formula to calculate $\left| {OA} \right|$.

$\left| {OA} \right| = \left| z \right| = \sqrt {{x^2} + {y^2}} $

Apply the modulus formula to calculate $\left| {OB} \right|$.

$\left| {OB} \right| = \left| {iz} \right| = \sqrt {{{\left( { - y} \right)}^2} + {x^2}} = \sqrt {{x^2} + {y^2}} = \left| z \right|$

Apply the area formula to calculate the area of the triangle

The area of the triangle is

$ = \dfrac{1}{2} \times \left| {OA} \right| \times \left| {OB} \right|$

$ = \dfrac{1}{2} \times \left| z \right| \times \left| z \right|$

$ = \dfrac{1}{2}{\left| z \right|^2}$

**Option ‘D’ is correct**

**Note:**If the angle between two complex numbers is $90^{\circ}$, then these two complex numbers are the legs of a triangle. So half of the product of the magnitude of the complex numbers is the area of the triangle that is made by these two complex numbers.

Last updated date: 07th Jun 2023

•

Total views: 91.2k

•

Views today: 0.48k

Recently Updated Pages

Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts

Name the Largest and the Smallest Cell in the Human Body ?

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Epipetalous and syngenesious stamens occur in aSolanaceae class 11 biology CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

A ball impinges directly on a similar ball at rest class 11 physics CBSE

Lysosomes are known as suicidal bags of cell why class 11 biology CBSE

Two balls are dropped from different heights at different class 11 physics CBSE

A 30 solution of H2O2 is marketed as 100 volume hydrogen class 11 chemistry JEE_Main

A sample of an ideal gas is expanded from 1dm3 to 3dm3 class 11 chemistry CBSE