Question

The area of the triangle whose vertices are (2,3) (2,4) (2,5) is A) O sq units B) 2 sq units C) 6 sq units D) 12 sq units ​

Answer 1

$correct \: answer \: is \: a) \: 0 \: sq. \: units$

REASON

$if \: we \: plot \: it \: on \: graph \: it \: will \: form \: a \: straght \: line \: as \: it \: cannot \: be \: triangle \\ \: thus \: 0 \: sq. \: units \: is \: area$

Answer 2

Given that , The three vertices of Triangle are : (2,3) , (2,4) , (2,5) .

Need To Find : Area of Triangle ?

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❍ Let's say that the three vertices of Triangle be A ( 2,3 ) , B ( 2 , 4 ) , C( 2 , 5 ) .

⠀⌬⠀We've three vertices of Triangle . Formula to Find Area of Triangle using Co – Ordinates is given by –

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

⠀⠀⠀⠀⠀Where ,

• x₁ = 2 , y₁ = 3
• x₂ = 2 , y₂ = 4
• x₃ = 2 , y₃ = 5

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

$:\implies \sf Area\:_{\:(\:\triangle\:\:ABC\:)\:}\:=\:\dfrac{1}{2}\:\Bigg[ \: x_1 \:\bigg(y_2 - y_3 \bigg) + \: x_2 \:\bigg(y_3 - y_1 \bigg) \:+\:\: x_3 \:\bigg(y_1 - y_2 \bigg) \:\Bigg]\:\\\\\\:\implies \sf Area\:_{\:(\:\triangle\:\:ABC\:)\:}\:=\:\dfrac{1}{2}\:\Bigg[ \: 2 \:\bigg( 4 - 5 \bigg) + \: 2 \:\bigg(5 - 3 \bigg) \:+\:\: 2 \:\bigg( 3 - 4 \bigg) \:\Bigg]\:\\\\\\:\implies \sf Area\:_{\:(\:\triangle\:\:ABC\:)\:}\:=\:\dfrac{1}{2}\:\Bigg[ \: 2 \:\bigg( -1 \bigg) + \: 2 \:\bigg( 2 \bigg) \:+\:\: 2 \:\bigg( -1 \bigg) \:\Bigg]\:\\\\\\:\implies \sf Area\:_{\:(\:\triangle\:\:ABC\:)\:}\:=\:\dfrac{1}{2}\:\Bigg[ \: \:\bigg( -2 \bigg) + \: \:\bigg( 4 \bigg) \:+\:\: \:\bigg( -2 \bigg) \:\Bigg]\:\\\\\\:\implies \sf Area\:_{\:(\:\triangle\:\:ABC\:)\:}\:=\:\dfrac{1}{2}\:\Bigg[ \: \: -2 + \: 4 \:\: -2 \:\Bigg]\:\\\\\\:\implies \sf Area\:_{\:(\:\triangle\:\:ABC\:)\:}\:=\:\dfrac{1}{2}\:\times \: \: 0 \:\:\\\\\\:\implies \underline {\boxed {\pmb{\frak{\purple { Area\:_{\:(\:\triangle\:\:ABC\:)\:}\:=\: \: \: 0 \:sq.units\:}}}}}\:\:\bigstar$

⠀⠀⠀⠀⠀As , The Final answer [ Area of Triangle ] is 0 sq.units which proves that , the given vertices are not of Triangle and they didn't form any Triangle .

$\\\therefore \:\underline {\sf Hence, \:The \:Area \:of \:Triangle \:is \:\:\pmb{\bf\: Option \:A\;)\: 0 \:sq.units \:}\:.}$