Question

the perimete tryr of a triangle is 450m . the ratio of i ints side's are 13:12:5 using herons formula find the area of the triangle​

Answer 1

GIVEN :-

• The perimeter of a triangle , s = 450 m.
• Ratio of sides of triangle are 13:12:5.

TO FIND :-

• The area of triangle by using heron's formula.

SOLUTION :-

Let the ratio constant be "x".

★ Sides of triangle = a = 13x , b = 12x , c =5x.

$\\ : \implies\displaystyle \sf \: Perimeter_{( \triangle)} \: = \dfrac{a + b + c}{2}$

$: \implies\displaystyle \sf \: 450 = \dfrac{13x + 12x + 5x}{2}$

$: \implies\displaystyle \sf \:450 = \dfrac{30x}{2}$

$: \implies\displaystyle \sf \:450 = 15x$

$: \implies\displaystyle \sf \:x = \frac{450}{15}$

$: \implies \underline{ \boxed{\displaystyle \sf \:x = 30m}}$

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$\\ \\ : \implies\displaystyle \sf \:side(a) = 13x$

$: \implies\displaystyle \sf \:side(a) = 13 \times 30 = \underline{\boxed{ \displaystyle \sf 390m}}$

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$\\ \\ : \implies\displaystyle \sf \:side(b) = 12x$

$: \implies\displaystyle \sf \:side(b) = 12 \times 30 = \underline{\boxed{ \displaystyle \sf 360m}}$

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$\\ \\ : \implies\displaystyle \sf \:side(c) = 5x$

$: \implies\displaystyle \sf \:side(c) = 5 \times 30= \underline{\boxed{ \displaystyle \sf 150m}}$

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

$\dashrightarrow \displaystyle \sf \: Area_{( \triangle)} = \sqrt{450(450 - 390)(450 - 360)(450 - 150)}$

$\dashrightarrow \displaystyle \sf \: Area_{( \triangle)} = \sqrt{40 \times 60 \times90 \times 300 }$

$\dashrightarrow \displaystyle \sf \: Area_{( \triangle)} = \sqrt{729000000}$

$\dashrightarrow \underline{ \boxed{\displaystyle \sf \: Area_{( \triangle)} = 27000 \: m ^{2} }}$

Answer 2

ANSWER✔

$\large\underline\bold{GIVEN,}$

$\dashrightarrow Perimeter\:of\: triangle= 450m$

$\dashrightarrow ratios\:of\:the\:sides\:of\:triangles\:are \\ \dashrightarrow 13:12:5$

✯FORMULA IN USE,

$\large{\boxed{\bf{ \star\:\: area\:of\:triangle_{(herons\:fomula)}= \sqrt{s(s - a)(s - b)(s - c)} \:\: \star}}}$

$\large\underline\bold{TO\:FIND,}$

$\dashrightarrow Area\:of\:triangle\:using\:herons\: formula.$

$\large\underline\bold{SOLUTION,}$

$\therefore let\:the\:constant\:be\:'x'\:m$

$\dashrightarrow sides\:of\:triangle\: are,\\ \dashrightarrow a=13x \\b= 12x \\ c= 5x$

$\therefore finding\:the\:value\:of\:x.$

$\dashrightarrow perimeter\:of\:triangle= a+b+c$

$\dashrightarrow 13x+12x+5x=450$

$\implies 30x=450$

$\implies x= \dfrac{450}{30}$

$\implies x = \cancel\dfrac{450}{30}$

$\implies x=15$

$\bf{\boxed{\sf{ \star\:\: x=15 \:\: \star}}}$

$\dashrightarrow x=15\\ a= 13x= 13\times 15= 195m\\ \dashrightarrow b=12x= 12\times 15 = 180m\\ \dashrightarrow c=5x= 5\times 15= 75m$

NOW, FINDING AREA OF TRIANGLE BY HERONS FORMULA,

$\therefore s= \dfrac{a+b+c}{2}$

$\implies s= \dfrac{195+180+75}{2}$

$\implies s= \dfrac{450}{2}$

$\implies s= 225$

$\bf\dashrightarrow area\:of\:triangle_{(herons\:fomula)}= \sqrt{s(s - a)(s - b)(s - c)}$

$\implies \sqrt{225(225-195)(225-180)(225-75)}$

$\implies \sqrt{225(30)(45)(150)}$

$\implies \sqrt{ 225(1350)(150)}$

$\implies 15\sqrt{202500}$

$\implies 15 \times 450$

$\implies 6750m^2$

$\large{\boxed{\bf{ \star\:\:area\:of\:triangle= 6750m^2 \:\: \star}}}$

$\large\underline\bold{AREA\:OF\:TRIANGLE\:IS\:6750m^2.}$

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