Question

$\begin{gathered}{\Huge{\textsf{\textbf{\underline{\underline{\color{green}{Question:}}}}}}}\end{gathered}$
The sides of a triangle are in the ratio of 12: 17: 25 and its perimeter is 540cm. Find its area.

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Answer 1

${\huge{ \colorbox{lightgreen}{\textsf{\textbf{\color{darkblue}{Required Answer»}}}}}}$

To Solve:

• Find the Area.

Given:

• Ratio of sides: 12:17:25
• Perimeter: 540 cm

Solⁿ:

• First find actual sides

• Find Semiperimeter

• Apply Heron's Formula

Ratios: 12x, 17x , 25x

$12x + 17x + 25x = 540 \\ \\ 54x = 540 \\ \\ x = 10 \\ \\ { \sf{ actual \: \: side = }} \\ \\ 12(10) = 120 \\ 17(10) = 170 \\ 25(10) = 250$

${ \sf{ now. \: \: semiperimeter}}$

${ \sf\frac{540}{2} = 270 }$

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

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

${ \sf{ \sqrt{270(150)(100)(20)}}}$

${ \sf \sqrt{3 \times 3 \times 3 \times 2 \times 5 \times 3 \times 5 \times 2 \times 5 \times 2 \times 5 \times 2 \times 5 \times 2 \times 2 \times 5}}$

${ \sf2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 5 \times 5 }$

${ \boxed{ \sf{ \purple{area = 9000 \: {cm}^{2}}}}}$

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Answer 2

$\huge\underline{\underline\bold \purple{solution}}$

$\huge\mathfrak{given - }$

$\bold{ \star \: ratio \: of \: the \: sides \: of \: the \: \triangle = 12 \ratio \: 17 \ratio \: 25}$

$\bold{\star \: perimeter = 540 \: cm}$

$\huge\mathfrak{find - }$

$\bigstar \: \bold{area}$

$\huge\mathfrak{steps - }$

$\sf \:\ddagger \: {assuming \: the \: ratios \: as - }$

$\dagger \: \purple{12x \ratio \: 17x \ratio \: 25x}$

$\sf \circ \: now \: we \: got \: the \: equation \: as -$

$\leadsto \bold{12x + 17x + 25x = 540} \\ \bold{\leadsto \: 54x = 540} \\ \bold{ \leadsto \: x = \frac{540}{54}} \\ \underline \bold{\leadsto \: x = 10}$

$\mapsto \sf \: 12x = 12 \times 10 = 120 \\ \mapsto \sf \: 17x = 17 \times 10 = 170 \\ \mapsto \sf \: 25x = 25 \times 10 = 250$

$\purple{\bold{ \longmapsto \: area \: of \: \triangle = \sqrt{s(s - a)(s - b)(s - c)} }}$

$\purple{\longmapsto \bold{semi \: perimeter = \frac{a + b + c}{2}}}$

$\star \: \bold{semi \: perimeter = \frac{540}{2} = 270 }$

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

$\leadsto \: area = \sqrt{270 \times 150 \times 100 \times 20}$

$\leadsto \: = \sqrt{18000000}$

$\purple{ \leadsto \: \fbox{ = 9000}}$

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