If z1 z2 z3 are vertices of an equilateral triangle

Asked by admin @ in Math viewed by 327 People

If Z1, Z2 ,Z3 are vertices of equilateral triangle then prove that Z1^2 + Z2^2 + Z3^2 = Z1.Z2 +Z2.Z3+ Z3.Z1.

If Zo is circumcenter of triangle then also prove that 3Zo^2 = Z1^2 + Z2^2 + Z3^2​

Answered by admin @


\huge {\underline {\underline{ \sf {ANSWER \: : }}}}

GIVEN :

\star \: \sf{Z_{1} ,Z_{2} ,Z_{3} \: are \: vertices \: of \:an \: equilateral \: triangle.}

TO PROVE :

\star\: \: \sf{Z_{1}}^{2} + {Z_{2}}^{2} + {Z_{3}}^{2} = Z_{1}.Z_{2} + Z_{2}. Z_{3} + Z_{3}.Z_{1}

\star \: \: \sf{ {3Z_{o}}^{2} = \: Z_{1}}^{2} + {Z_{2}}^{2} + {Z_{3}}^{2}

SOLUTION :

  • Let us consider \sf{Z_{1} ,Z_{2} ,Z_{3}} to be the vertices of the equilateral triangle.

We know that, each angle of the equilateral triangle is 60° i.e, \sf \frac{ \pi}{3}.

We have the formula, when the complex number is not rotating about its origin which is,

\longrightarrow \: \boxed{ \sf{ \:\blue{\frac{Z_{3} -Z_{1}} { \mid \:{Z_{3} -Z_{1}} \mid} =\frac{Z_{2} -Z_{1}} { \mid \:{Z_{2} -Z_{1}} \mid \: \: } {e}^{i \alpha } }}}

For, Anticlockwise Direction

\implies \: \sf{ \: \frac{Z_{3} -Z_{1}} { \mid \:{Z_{3} -Z_{1}} \mid} =\frac{Z_{2} -Z_{1}} { \mid \:{Z_{2} -Z_{1}} \mid \: \: } {e}^{i \frac{ \pi}{3} } }

As, there the triangle is equilateral, therefore \sf \mid \:{Z_{3} - Z_{1}} \mid = \mid \:{Z_{2} - Z_{1}} \mid = l(lenght \: of\: the\: sides)

\implies \: \sf{ \: \frac{Z_{3} -Z_{1}} { \cancel{l}} =\frac{Z_{2} -Z_{1}} { \cancel{l}} {e}^{i \frac{ \pi}{3} } }

\implies \: \sf{ \: {Z_{3} -Z_{1}} = (\: Z_{2} -Z_{1} \:) {e}^{i \frac{ \pi}{3}}} \: \: \: \: \: \longrightarrow \: (1)

____________________________________

Now, for Clockwise Direction

\implies \: \sf{ \: \frac{Z_{3} -Z_{2}} { \mid \:{Z_{3} -Z_{2}} \mid} =\frac{Z_{1} -Z_{2}} { \mid \:{Z_{1} -Z_{2}} \mid \: \: } {e}^{ - i \frac{ \pi}{3}}}

Here also, \sf \mid \:{Z_{3} - Z_{1}} \mid = \mid \:{Z_{2} - Z_{1}} \mid = l(lenght \: of\: the\: sides)

\implies \: \sf{ \: \frac{Z_{3} -Z_{2}} { \cancel{l}} =\frac{Z_{1} -Z_{2}} { \cancel{l}} {e}^{ - i \frac{ \pi}{3} } }

\implies \: \sf{ \: {Z_{3} -Z_{2}} = (\: Z_{1} -Z_{2} \:) {e}^{ - i \frac{ \pi}{3}}} \: \: \: \: \: \longrightarrow \: (2)

____________________________________

Now, Multiplying (1) and (2) we get,

\implies \: \sf{ \:( {Z_{3} -Z_{1}})({Z_{3} -Z_{2}}) = (\: Z_{2} -Z_{1} \:).(\: Z_{1} -Z_{2} \:) \: {e}^{i \frac{ \pi}{3}}. {e}^{ - i \frac{ \pi}{3}}}

\implies \: \sf{ \:( {Z_{3} -Z_{1}})({Z_{3} -Z_{2}}) = (\: Z_{2} -Z_{1} \:).(\: Z_{1} -Z_{2} \:) \: {e}^{i \frac{ \pi}{3} - i \frac{ \pi}{3} } } \:

\implies \: \sf{ \:( {Z_{3} -Z_{1}})({Z_{3} -Z_{2}}) = - {(\: Z_{1} -Z_{2} )}^{2} \: {e}^{0}}

\sf \: \implies \: {Z_{3}}^{2} - Z_{2}.Z_{3} - Z_{1}.Z_{3} + Z_{1}.Z_{2} = - {Z_{1}}^{2} - {Z_{2}}^{2} + 2Z_{1}.Z_{2}

\sf \: \: \boxed { \: \green{ \therefore \: \: \sf {Z_{1}}^{2} + {Z_{2}}^{2} + \: {Z_{3}}^{2} = Z_{1}.Z_{2} + Z_{2}.Z_{3} + Z_{3}.Z_{1}}}

Hence Proved .

____________________________________

Now, For Equilateral Triangle,

As Z_{o} is the Circumcentre of the triangle and also the centroid as the triangle is equilateral.

\sf \therefore Z_{o} = \frac{Z_{1} + Z_{2} + Z_{3} }{3}

\sf \implies 3Z_{o} = Z_{1} + Z_{2} + Z_{3}

Squaring both sides we get,

\sf \: \implies {(3Z_{o}) }^{2} = ({Z_{1} + Z_{2} + Z_{3})}^{2}

\sf \implies {9Z_{o} }^{2} = {Z_{1} } ^{2} + {Z_{2}}^{2} + \: {Z_{3}}^{2} + 2Z_{1}.Z_{2} + 2Z_{2}.Z_{3} + 2Z_{3}.Z_{1}

(Replacing, \sf 2Z_{1}.Z_{2} + 2Z_{2}.Z_{3} + 2Z_{3}.Z_{1} by \sf {Z_{1}}^{2} + {Z_{2}}^{2} + \: {Z_{3}}^{2} , as we proved earlier we get)

\sf \implies {9Z_{o} }^{2} = 3{Z_{1} } ^{2} + 3{Z_{2}}^{2} + \: 3{Z_{3}}^{2}

\sf \implies {3Z_{o} }^{2} = {Z_{1} } ^{2} + {Z_{2}}^{2} + \: {Z_{3}}^{2}

\sf {\therefore {\boxed {\green {{3Z_{o} }^{2} = {Z_{1} } ^{2} + {Z_{2}}^{2} + \: {Z_{3}}^{2} } } }}

Hence Proved.


Similar Questions

If z1 z2 z3 are complex numbers such that

Asked by admin @ in Math viewed by 271 persons

If z1,z2 and z3 are complex numbers such that |z1| = |z2| = |z3| = |1/z1 + 1/z2 + 1/z3| = 1, then |z1 + z2 + z3| is

Find the area of the triangle with vertices

Asked by admin @ in Math viewed by 324 persons

Determine the area of the triangle with vertices A(1, 1,2) B (2,4,6) C(0,5,5)​

Find the area of triangle whose vertices are

Asked by admin @ in Math viewed by 294 persons

Find the area of the triangle whose vertices are (2,3),(-1,0),(2,-4)full step needed

Find the circumradius of a triangle whose vertices are

Asked by admin @ in Math viewed by 280 persons

find the coordinates of the circumcentre of the triangle whose vertices (8,6) , (8,-2) & (2,-2). also find the circumradius.

If the vertices of a triangle have integral coordinates

Asked by admin @ in Math viewed by 344 persons

if the vertices of a triangle have integral coordinates, prove that the triangle cannot be equilateral.

The vertices of triangle abc are a 5 5

Asked by admin @ in Math viewed by 297 persons

The vertices of a triangle ABC are A(5,2),B(-9, -3) and C(-3,-5). D, E, F are respectively the mid points of BC, CA and AB. Prove that the area ofA …

In a triangle abc with vertices a 2 3

Asked by admin @ in Math viewed by 345 persons

Question 17 In the triangle ABC with vertices A (2, 3), B (4, –1) and C (1, 2), find the equation and length of altitude from the vertex A. Class …

Find the orthocentre of a triangle whose vertices are

Asked by admin @ in Math viewed by 314 persons

Find the coordinates of the orthocentre of the triangle whose vertices are (1,2) ,(2,3) and (4,3).

Find the area of a triangle whose vertices are

Asked by admin @ in Math viewed by 337 persons

Find the area of the triangle whose vertices are (2,3),(-1,0),(2,-4)full step needed

Locus of centroid of the triangle whose vertices are

Asked by admin @ in Math viewed by 422 persons

locus of centroid of the triangle whose vertices are (acost,asint),(bsint,-bcost)and(1,0),where t is the parameter , is

Find the centroid of the triangle whose vertices are

Asked by admin @ in Math viewed by 331 persons

Find the centroid of the triangle whose vertices are: (3, -5), (-7, 4), (10, -2)

In the triangle abc with vertices a 2 3

Asked by admin @ in Math viewed by 346 persons

Question 17 In the triangle ABC with vertices A (2, 3), B (4, –1) and C (1, 2), find the equation and length of altitude from the vertex A. Class …

Find the centroid of a triangle whose vertices are

Asked by admin @ in Math viewed by 386 persons

Find the centroid of the triangle whose vertices are: (3, -5), (-7, 4), (10, -2)

Find the area of the triangle whose vertices are

Asked by admin @ in Math viewed by 303 persons

Find the area of the triangle whose vertices are (2,3),(-1,0),(2,-4)full step needed

Find the area of triangle whose vertices are 1 1

Asked by admin @ in Math viewed by 281 persons

Calculate the area of the triangle whose vertices are (1,1) , (2,3) ,(4,5)

Most viewed questions in Math

What is the value of x in the expression

Asked by admin @ in Math viewed by 15126 persons

Shubham is facing south and moves 30 km

Asked by admin @ in Math viewed by 12977 persons

Prachi excellence in mathematics class 7 solutions pdf free download

Asked by admin @ in Math viewed by 12021 persons

Oxford new enjoying mathematics class 7 solutions chapter 1

Asked by admin @ in Math viewed by 11514 persons

Survey of various types of bank accounts icse project

Asked by admin @ in Math viewed by 11501 persons

The town of p is located at point x

Asked by admin @ in Math viewed by 11355 persons

A clock gains 4 minutes after every 4 minutes

Asked by admin @ in Math viewed by 10161 persons

Running a tuck shop or canteen maths project pdf

Asked by admin @ in Math viewed by 9375 persons

Paul is sixteenth from the front of the row

Asked by admin @ in Math viewed by 9345 persons

The average weight of 20 teachers is 80 kg

Asked by admin @ in Math viewed by 7517 persons

The number 567 xy is completely divisible by 30

Asked by admin @ in Math viewed by 7393 persons

In a row of friends tiya occupies fifteenth place

Asked by admin @ in Math viewed by 7200 persons

What would $ mean if fat is coded as

Asked by admin @ in Math viewed by 5271 persons

Consider a 3 digit integer x with distinct digits

Asked by admin @ in Math viewed by 4797 persons

Rahul went to his mother's mother in law

Asked by admin @ in Math viewed by 4174 persons