Question

find the area of the triangle whose vertices are (-3,2),(-1,-1) and (1,2)

Answer 1

Given that , The three vertices ( or corners ) of Triangle are ( -3 , 2 ) , ( -1 , -1 ) and ( 1 , 2 ) .

Exigency To Find : The Area of Triangle ?

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❍ Let's say that the three vertices of Triangle be D ( -3 , 2 ) , E ( -1 , -1 ) and F ( 1 , 2 ) .

⠀⠀⠀▪︎ We've been provided with three vertices of Triangle [ i.e. D ( -3 , 2 ) , E ( -1 , -1 ) and F ( 1 , 2 ) ] and we have to find out the Area of Triangle using the given Co – Ordinates of Vertices of Triangle .

》 Formula to Calculate Area of Triangle using Co – Ordinates is given by :

$\qquad \star \:\:\underline{\boxed{\frak{ Area\:_{\:(Triangle)\:}\:=\:\dfrac{1}{2}\:\Bigg[ \: x_1 \:\bigg(y_2 - y_3 \bigg) + \: x_2 \:\bigg(y_3 - y_1 \bigg) \:+\:\: x_3 \:\bigg(y_1 - y_2 \bigg) \:\Bigg]\:sq.units \:}}}$

$\frak{ Where \:}\begin{cases} \sf \:\:x_1\:=\:\frak{ -3}\:\:\& \:\:\:y_1\:=\:\frak{ 2}\: \\ \sf \:\:x_2\:=\:\frak{ -1}\:\:\& \:\:\:y_2\:=\:\frak{ -1}\: \\\sf \:\:x_3\:=\:\frak{ 1}\:\:\:\:\& \:\:\:y_3\:=\:\frak{ 2}\: \end{cases}\:$

$\qquad \dag \:\underline {\frak{Substituting \:known \:Values \:in \:Formula \:\::\:}}$

$\qquad:\implies \sf Area\:_{\:(Triangle)\:}\:=\:\dfrac{1}{2}\:\Bigg[ \: x_1 \:\bigg(y_2 - y_3 \bigg) + \: x_2 \:\bigg(y_3 - y_1 \bigg) \:+\:\: x_3 \:\bigg(y_1 - y_2 \bigg) \:\Bigg]\:$

$\qquad:\implies \sf Area\:_{\:(\triangle\: DEF)\:}\:=\:\dfrac{1}{2}\:\Bigg[ \: -3 \:\bigg( \{ - 1 \} - 2 \bigg) + \: ( -1 ) \:\bigg( 2 - \{ - 3 \} \bigg) \:+\:\: 1 \:\bigg( \{ - 3 \} - \{ - 1 \} \bigg) \:\Bigg]\:$

$\qquad:\implies \sf Area\:_{\:(\triangle\:DEF)\:}\:=\:\dfrac{1}{2}\:\Bigg[ \: -3 \:\bigg( - 1 - 2 \bigg) + \: (-1 )\:\bigg( 2 + 3 \bigg) \:+\:\: 1 \:\bigg( - 3 + 1 \bigg) \:\Bigg]\:$

$\qquad:\implies \sf Area\:_{\:(\triangle DEF )\:}\:=\:\dfrac{1}{2}\:\Bigg[ \: -3 \:\bigg( -3 \bigg) - \: 1 \:\bigg( 5 \bigg) \:+\:\: 1 \:\bigg( - 2 \bigg) \:\Bigg]\:$

$\qquad:\implies \sf Area\:_{\:(\triangle\:DEF)\:}\:=\:\dfrac{1}{2}\:\Bigg[ \: \:\bigg( 9 \bigg) + \: \:\bigg( - 5 \bigg) \:+\:\: \:\bigg( -2 \bigg) \:\Bigg]\:$

$\qquad:\implies \sf Area\:_{\:(\triangle\:DEF)\:}\:=\:\dfrac{1}{2}\:\Bigg[ \: \: 9 \:- \: 5 \:-\:\: \:2 \:\Bigg]\:$

$\qquad:\implies \sf Area\:_{\:(\triangle\:DEF)\:}\:=\:\dfrac{1}{2}\:\Bigg[ \: \: 4 \:-\:\: \:2 \:\Bigg]\:\\\\\qquad:\implies \sf Area\:_{\:(\triangle\:DEF)\:}\:=\:\dfrac{1}{2}\:\times \: \: 2 \:\:$

$\qquad:\implies \underline {\boxed {\pmb{\frak{\purple {\:\:Area\:_{\:(\triangle\:DEF)\:}\:=\:1 \:sq.units \:}}}}}\:\:\bigstar \:\:$

∴ Hence , Area of Triangle is 1 sq.units .