Question

5. Sides of a triangle are in the ratio of 12:17:25 and its perimeter is 540cm. find its area using heron formula​

Answer 1

$\bf \underline{ \underline{\maltese\:Given} }$

$\sf \Rrightarrow Sides \: of \: a \: triangle \: are \: in \: the \: ratio \: of \: 12:17:25$

$\sf \Rrightarrow Perimeter \: of \: the \: triangle = 540 \: cm$

$\bf \underline{ \underline{\maltese\:To \: find } }$

$\sf \Rrightarrow Area \: of \: triangle = \: ? \: \: \: \rm [By \: using \: Heron's \: formula]$

$\bf \underline{ \underline{\maltese\:Solution } }$

$\sf Let \: the \: sides \: of \: the \: triangle \: be :$

$\sf \implies First \: side \: (a) = 12x$

$\sf \implies Second \: side \: (b) = 17x$

$\sf \implies Third \: side \: (c) = 25x$

$\bf \underline{ Now},$

$\sf \implies 12x+ 17x+ 25x = 540$

$\sf \implies 54x = 540$

$\sf \implies x = \cancel \dfrac{540}{54}$

$\sf \implies x = 10$

$\sf Sides \: of \: triangle \: are :$

$\sf \implies First \: side \: (a) = 12x = \bf12 \times 10 = 120 \: cm$

$\sf \implies Second \: side \: (b) = 17x = \bf 17 \times 10 = 170 \: cm$

$\sf \implies Third \: side \: (c) = 25x = \bf 25 \times 10 = 250 \: cm$

$\bf \underline{ Now},$

$\sf Area \: of \: \bf \triangle \sf\: using \: Heron's \: formula :$

$\underline{\boxed{\sf{Area_{\triangle} = \sqrt{s(s - a)(s - b)(s - c)}}}}$

$\sf \implies s \: = \: \dfrac{a + b + c}{2}$

$\sf \implies s \: = \: \dfrac{120 + 170 + 250}{2}$

$\sf \implies s \: = \: \cancel\dfrac{540}{2}$

$\sf \implies s \: = \: 270$

$\sf \underline{Thus, \: Semi-perimeter \: is \: 270 \: cm }$

$\bf \underline{Now}, \sf \: area \: of \: triangle :$

$\sf{Area_{\triangle} = \sqrt{s(s - a)(s - b)(s - c)}}$

$\sf{Area = \sqrt{270(270 - 120)(270 - 170)(270 - 250)}}$

$\sf{Area = \sqrt{270 \: ( 150 \times 100 \times 20)}}$

$\sf{Area = \sqrt{270 \times 300000}}$

$\sf{Area = \sqrt{81000000}}$

$\sf Rewrite \: 81000000 \: as \: 9000^2$

$\sf{Area = \sqrt{9000 ^{2} }}$

$\sf{Area = 9000 \: cm^{2} }$

$\underline{ \boxed{ \bf \red{Thus, \: area \: of \: triangle\: is\: 9000 \: cm ^{2}}}}$

Answer 2

Given,

• ratio of sides of triangle is 12:17:25. And, Perimeter of triangle is 540 cm.

Let's consider the sides of triangles a, b and c be 12x, 17x and 25x.

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We know that,

• Perimeter of a triangle is the sum of its sides. Then,

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$\begin{gathered}\qquad\qquad\dashrightarrow\sf 12x + 17x + 25x = 540\\\\\\ \qquad\qquad\dashrightarrow\sf 54x = 540\\\\\\ \qquad\qquad\dashrightarrow\sf x = \cancel{\dfrac{540}{54}}\\\\\\ \qquad\qquad\dashrightarrow\sf {\underline{\boxed{\pmb{\frak{x = 10}}}}}\:\bigstar\\\\\end{gathered}$

∴ Sides of triangle will be 120 cm, 170 cm & 250 cm.

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If Perimeter of triangle is 540 cm. Then, Semi-perimeter (s) will be 270 cm.

If Perimeter of triangle is 540 cm. Then, Semi-perimeter (s) will be 270 cm.⠀⠀⠀⠀

$\begin{gathered}\bigstar\:{\underline{\sf{Using\:Heron's\:formula\:to\:find\:Area\:of\:\triangle\::}}}\\\\\end{gathered}$

$\begin{gathered}\bigstar\:{\underline{\boxed{\pmb{\sf{Area_{\:(triangle)} = \sqrt{s\bigg(s - a\bigg)\bigg(s - b\bigg)\bigg(s - c\bigg)}}}}}}\\\\\end{gathered}$

$\begin{gathered}\dashrightarrow\sf \sqrt{s\bigg(270 - 120\bigg)\bigg(270 - 170\bigg)\bigg(270 - 250\bigg)} \\\\\\ \dashrightarrow\sf \sqrt{250 \times 150 \times 100 \times 20}\\\\\\ \dashrightarrow\sf {\underline{\boxed{\pmb{\frak{\pink{9000\:cm^2}}}}}}\:\bigstar\\\\\end{gathered}$

$\therefore\:{\underline{\sf{Area\:of\:triangle\:is\:{\pmb{9000\:cm^2}}}}}$