Math, asked by Isha077, 9 months ago

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​

Answers

Answered by pal69
18

Answer:

Since z1, z2, z3, are vertical of an equilateral triangle

therefore circcumcenter (z0) =centroid

(z1+z2+z3) /3...........1)

also for an equilateral triangle

z²1+z²2+z²3=z1z2+z2z3+z3z1........ 2)

on squaring (1). we get,

9z²0=z²1+z²2+z²3+2(z1z2+z2z3+z3z1)

9z²0=z²1+z²2+z²3+2(z²1+z²2+z²3)

3z²0=z²1+z²2+z²3

Answered by BraɪnlyRoмan
363

 \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.


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