Question

Find fourth vertex of the parallelogram whose three vertices are A( 3, -4) B ( -1, -3) C( -6, 2).

Answer 1

Given : The three vertices of Parallelogram are A( 3, -4) B ( -1, -3) C( -6, 2) .

Exigency to find : The Fourth Vertex or Vertex of D of Parallelogram.

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❍ Let's Consider the Fourth Vertex or Vertex of D of Parallelogram be ( x , y ) .

As , we know that ,

• In Parallelogram the Diagonal bisects each other .

Therefore ,

⠀⠀━━ Midpoint of AC = Midpoint of BD . [ Refer The given attachment ]

$\dag\:\:\it{ As,\:We\:know\:that\::}$

$\qquad \dag\:\:\bigg\lgroup \sf{Mid\:Point \:: \bigg( \dfrac{ x_1 + x_2 }{2} , \dfrac{y_1 + y_2 }{2} \bigg) }\bigg\rgroup$

⠀⠀━━Now , By Using Mid Point Theorem :

⠀⠀━━ Midpoint of AC = Midpoint of BD .

$\qquad :\implies \sf \bigg( \dfrac{ 3 -6 }{2} , \dfrac{ -4 + 2}{2} \bigg) = \bigg( \dfrac{ -1 + x }{2} , \dfrac{ -3 + y}{2} \bigg) \\\\ :\implies \sf \bigg( \dfrac{ -3 }{2} , \dfrac{ -2}{2} \bigg) = \bigg( \dfrac{ -1 + x }{2} , \dfrac{ -3 + y}{2} \bigg) \\\\ \bf Therefore\:\: : \\\\ :\implies \sf \bigg( \dfrac{ -3 }{2} = \dfrac{ -1 + x }{2} \bigg) , \bigg( \dfrac{ -2 }{2} = \dfrac{ -3 + y}{2} \bigg)$

⠀⠀⠀⠀⠀Finding x :

$\qquad:\implies \sf \bigg( \dfrac{ -3 }{2} = \dfrac{ -1 + x }{2} \bigg)$

⠀By Cross Multiplication :

$\qquad:\implies \sf \bigg( \dfrac{ -3 }{2} = \dfrac{ -1 + x }{2} \bigg)$

$\qquad:\implies \sf \bigg( 2(-3) = 2( -1 + x) \bigg)$

$\qquad:\implies \sf \bigg( -6 = 2( -1 + x) \bigg)$

$\qquad:\implies \sf \bigg( -6 = -2 + 2x \bigg)$

$\qquad:\implies \sf \bigg( -6 + 2 = 2x \bigg)$

$\qquad:\implies \sf \bigg( -4 = 2x \bigg)$

$\qquad:\implies \sf \bigg( \cancel {\dfrac{-4}{2}} = x \bigg)$

$\qquad \longmapsto \frak{\underline{\purple{\:x = -2 }} }\bigstar$

⠀⠀⠀⠀⠀Finding y :

$\qquad:\implies \sf \bigg( \dfrac{ -2 }{2} = \dfrac{ -3 + y}{2} \bigg)$

⠀⠀By Cross Multiplication :

$\qquad:\implies \sf\bigg( \dfrac{ -2 }{2} = \dfrac{ -3 + y}{2} \bigg)$

$\qquad:\implies \sf\bigg( 2(-2) = 2( -3 + y) \bigg)$

$\qquad:\implies \sf\bigg( -4 = 2( -3 + y) \bigg)$

$\qquad:\implies \sf\bigg( -4 = -6 + 2y \bigg)$

$\qquad:\implies \sf\bigg( -4 + 6 = 2y \bigg)$

$\qquad:\implies \sf\bigg( 2 = 2y \bigg)$

$\qquad:\implies \sf \bigg( \cancel {\dfrac{2}{2}} = y \bigg)$

$\qquad \longmapsto \frak{\underline{\purple{\:y = 1 }} }\bigstar$

Therefore,

• x = -2 .
• y = 1 .

Therefore,

⠀⠀⠀⠀⠀$\therefore {\underline{ \mathrm {\:The\:Vertex\:D\:or\:Fourth\:Vertex \:of\:Parallelogram \:is\:\bf{(-2,1)}}}}$

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