Question

The area of a triangular field is
1344 sq metre . If the sides of the field are in the
ratio 4: 7: 9 , find the sides of the field . ( Take √5 = 2.24)

Answer 1

$\large\underline{\underline{☢\color{maroon}{\pmb{\sf{\:Given :-}}}}}$

• ➬ Area of triangle = 1344 m²
• ➬ Ratios of sides = 4:7:9

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$\large\underline{\underline{☢\color{maroon}{\pmb{\sf{\:To \: Find :-}}}}}$

• ➬ Sides of the triangle = ?

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$\large\underline{\underline{☢\color{maroon}{\pmb{\sf{\:Solution :-}}}}}$

❒ We know that :

$\large\green\bigstar{\underline{\boxed{\color{blue}{\sf{Area{\small{(Triangle)}} = {\pink{\sf{ \sqrt{s(s - a)(s - b)(s - c)} }} }}}}}}$

❒ Let the ratios :

• ➳ Area = 1344 m²
• ➳ S = Semi - perimeter = ?
• ➳ 1st side = 4x
• ➳ 2nd side = 7x
• ➳ 3rd side = 9x

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:★━━━━━━━━━━━━━━━━━━★$

❒ Finding the S :

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

${:\implies{\sf{ \: \: \: \: \: \: \: \: \: \: \: \: \dfrac{4x + 7x+ 9x}{2}}}}$

${:\implies{\sf{ \: \: \: \: \: \: \: \: \: \: \: \: \dfrac{\cancel{20}x}{\cancel2}}}}$

$\large{\blue{:\leadsto{\underline{\sf{ S = \color{cyan} 10x}}}}}$

❒ Applying the values in formula :

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

${:\implies{\sf{ 1344 = \sqrt{10x(10x - 4x)(10x- 7x)(10x- 9x)} }}}$

${:\implies{\sf{ 1344 = \sqrt{10x \times 6x \times 3x \times 1x} }}}$

${:\implies{\sf{ 1344 = \sqrt{5x \times 2x\times 3x \times 2x \times 3x \times 1x} }}}$

${:\implies{\sf{ 1344 = \sqrt{5x \times \underline{2x\times 2x }\times \underline{3x \times 3x }\times 1x} }}}$

${:\implies{\sf{ 1344 = \sqrt{5x \times 2x \times 3x} }}}$

${:\implies{\sf{ 1344 = \sqrt{2.24x\times 6x} }}}$

${:\implies{\sf{ 1344 = 13.44x }}}$

${:\implies{\sf{ x = \cancel\frac{1344}{13.44} }}}$

$\large{\green{:\leadsto{\underline{\sf{ X= \color{red} 10}}}}}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \:★━━━━━━━━━━━━━━━━━━★$

❒ Sides of triangle :

${\large\orange{:\longmapsto{\sf{1st\: Side = 4x = 4 \times 10 = 40m}}}}$

${\large\orange{:\longmapsto{\sf{2nd\: Side = 7x = 7\times 10 = 70m}}}}$

${\large\orange{:\longmapsto{\sf{3rd\: Side = 9x = 9\times 10 = 90m}}}}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \:★━━━━━━━━━━━━━━━━━━★$

❒ Verification :

${\twoheadrightarrow{\sf{\sqrt {s(s - a)(s - b)(s - c) }= 1344 \: {m}^{2} }}}$

${\twoheadrightarrow{\sf{\sqrt {10x(10x - 4x)(10x- 7x)(10x - 9x) }= 1344 \: {m}^{2} }}}$

${\twoheadrightarrow{\sf{\sqrt {10 \times 10(10 \times 10- 4 \times 10)(10 \times 10- 7 \times 10)(10 \times 10- 9 \times 10) }= 1344 \: {m}^{2} }}}$

${\twoheadrightarrow{\sf{\sqrt {100(100- 40)(100- 7 0)(100- 90) }= 1344 \: {m}^{2} }}}$

${\twoheadrightarrow{\sf{\sqrt {100 \times 60 \times 30 \times 10 }= 1344 \: {m}^{2} }}}$

${\twoheadrightarrow{\sf{\sqrt {5 \times 5 \times 2 \times 2 \times 2 \times 3 \times \sqrt{5} }= 1344 \: {m}^{2} }}}$

${\twoheadrightarrow{\sf{\sqrt {25 \times 4 \times 6 \times \sqrt{5} }= 1344 \: {m}^{2} }}}$

${\twoheadrightarrow{\sf{\sqrt {25 \times24 \times 2.24}= 1344 \: {m}^{2} }}}$

${\twoheadrightarrow{\sf{1344 \: {m}^{2} = 1344 \: {m}^{2} }}}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \large{\red{\underline{\bf{LHS = RHS}}}}$

Hence, Verified .

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✬Know more :

❒ Formulas for Area :

$\begin{gathered}\red{\large \qquad \boxed{\boxed{\begin{array}{cc} \ \color{blue}➳ \: \: \bf Square = Side² \\ \\ \ \color{orange}➳ \: \: \bf Rectangle = Length \times Breadth \\ \\ \ \color{green}➳ \: \: \bf Triangle = \sqrt{s(s - a)(s - b)(s - c)}\end{array}}}}\end{gathered}$

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