Question

Find the area of a triangle two sides of which are 18cm and 10cm and the perimeter is 42cm


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Answer 1

Given : The two sides of Triangle are 18 cm and 10 cm and Perimeter of Triangle is 42 cm .

Need To Find : Area of Triangle ?

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❍ Let's Consider the third side of Triangle be x cm .

$\dag{\underline {\frak{ As \:We \:know \:that \: \::\:}}}$

⠀⠀⠀★ The Sum of all sides of Triangle is the Perimeter of Triangle & Perimeter of a given Triangle is 42 cm .

$\qquad \twoheadrightarrow \:\pmb{\sf \: Perimeter \:_{\:( Triangle)\:}\:=\: a\:+\:b\:+\:c\:}\\\\\\ \qquad \twoheadrightarrow \:\sf 18 \:+\:10\:+\:x \:=\: 42 \\\\\\ \qquad \twoheadrightarrow \:\sf 28\:+\:x \:=\: 42 \\\\\\ \qquad \twoheadrightarrow \:\sf \:x \:=\: 42\:-\:28 \\\\\\ \qquad \twoheadrightarrow \:\pmb{\underline {\boxed {\purple {\frak{ \:x \:=\: 14\:cm \:}}}}}\:\:\bigstar$

∴ Hence , Three Sides of Triangle are 18 cm , 10 cm & 14 cm .

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★ CalCuLaTing ArEa oF TriAngLE :

$\qquad \star \pmb{\underline {\boxed {\frak{ Area_{\:(Triangle)\:}\:=\: \sqrt{\: s\: \bigg( s - a \bigg) \:\bigg( s - b \bigg)\:\bigg( s - c \bigg)\:}\:}}}}$

$\frak{ Here}\:\begin{cases}\quad \:\sf a \: =\: \frak{\pmb{18\:cm \:}} \\ \quad \:\sf b \: =\: \frak{\pmb{10\:cm \:}} \\\quad \:\sf c \: =\: \frak{\pmb{14\:cm \:}} \\ \quad \:\sf s \: =\: \frak{\pmb{21\:cm \:}}\:i.e,\:Semi\:-\:Perimeter\:(s)\:=\:\dfrac{42}{2}\:=\:21\:cm\end{cases}$

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

$:\implies \sf Area_{\:(Triangle)\:}\:=\: \sqrt{\: s\: \bigg( s - a \bigg) \:\bigg( s - b \bigg)\:\bigg( s - c \bigg)\:}\:$

$:\implies \sf Area_{\:(Triangle)\:}\:=\: \sqrt{\: 21\: \bigg( 21 - 18 \bigg) \:\bigg( 21 - 10 \bigg)\:\bigg( 21 - 14 \bigg)\:}\:$

$:\implies \sf Area_{\:(Triangle)\:}\:=\: \sqrt{\: 21\: \bigg( 3 \bigg) \:\bigg( 11 \bigg)\:\bigg( 7 \bigg)\:}\:$

$:\implies \sf Area_{\:(Triangle)\:}\:=\: \sqrt{\: 21\: \times 3 \times \:\times 11 \times\: 7 \:}\:$

$:\implies \sf Area_{\:(Triangle)\:}\:=\: \sqrt{\: 21\: \times 21 \times 11 \:}\:$

$:\implies \pmb{\underline {\boxed {\frak{\purple { Area_{\:(Triangle)\:}\:=\: 21\sqrt{\: 11 \:}\:cm^2\:}}}}}\:\:\bigstar$