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​ by​

Answer 1

$\huge\mathtt{\fbox{\red{Answer✍︎}}}$

$\large\underline\mathfrak{\pink{GIVEN,}}$

$\dashrightarrow \red{ Perimeter\:of\: triangle= 450m}$

$\dashrightarrow \orange{ratios\:of\:the\:sides\:of\:triangles\:are }\\ \dashrightarrow \pink{13:12:5}$

$\large{\boxed{\bf{ \mathfrak{\blue{FORMULA,}}}}}$

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

$\large\underline\mathfrak{\pink{TO\:FIND,}}$

$\dashrightarrow \green{Area\:of\:triangle\:using\:herons\: formula.}$

$\large\underline\mathfrak{\purple{SOLUTION,}}$

$\therefore \green{let\:the\:constant\:be\:'x'\:m}$

$\dashrightarrow \red{sides\:of\:triangle\: are,}\\ \dashrightarrow \blue{a=13x} \\ \purple{b= 12x }\\ \green{c= 5x }$

$\therefore \orange{finding\:the\:value\:of\:x.}$

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

$\dashrightarrow \blue{13x+12x+5x=450}$

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

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

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

$\implies \green{x=15}$

$\rm{\boxed{\bf{ \:\: x=15 \:\: }}}$

$\dashrightarrow \red{x=15}\\ \pink{a= 13x= 13\times 15= 195m}\\ \blue{\dashrightarrow b=12x= 12\times 15 = 180m}\\ \dashrightarrow \purple{c=5x= 5\times 15= 75m}$

NOW, FINDING AREA OF TRIANGLE BY HERONS FORMULA,

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

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

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

$\implies \orange{s= 225}$

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

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

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

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

$\implies \purple{15\sqrt{202500}}$

$\implies \purple{15 \times 450}$

$\implies \purple{6750m^2}$

$\rm{\boxed{\sf{ \large{\circ}\:\:area\:of\:triangle= 6750m^2 \:\: \large{\circ}}}}$

$\rm\underline\mathfrak{\pink{AREA\:OF\:TRIANGLE\:IS\:6750m^2.}}$

Answer 2

GivEn:

• The Perimeter of a triangle is 450 m.
• The ratio of sides of triangle are 13:12:15.

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To find:

• Area of triangle using Heron's formula.

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Solution:

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☯ Let sides of triangle be 13x, 12x and 5x.

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$\sf where \begin{cases} & \sf{a = \bf{13\;x}} \\ & \sf{b = \bf{12\;x}} \\ & \sf{c = \bf{5\;x}} \end{cases}$

$\underline{\bigstar\:\boldsymbol{According\:to\:the\:question\::}}$

• The Perimeter of a triangle is 450 m.

⠀⠀⠀⠀⠀⠀⠀

$:\implies\sf 13x + 12x + 5x = 450$

$:\implies\sf 30x = 450$

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

$:\implies{\boxed{\frak{\pink{x = 15}}}}\;\bigstar$

⠀⠀━━━━━━━━━━━━━━━━━━━━━

Therefore,

⠀⠀⠀⠀⠀⠀⠀

• a = 13x = 195 m

• b = 12x = 180 m

• c = 5x = 75 m

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${\underline{\sf{\bigstar\;Using\; Heron's\;Formula\;:}}}$

We know that,

⠀⠀⠀⠀⠀⠀⠀

$\rightarrow\sf s = \dfrac{a + b + c}{2}\qquad\qquad\bigg\lgroup\bf s = semi - perimeter \bigg\rgroup$

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

$\rightarrow\sf s = \cancel{ \dfrac{450}{2}}$

$\rightarrow\bf \red{s = 225}$

Now,

⠀⠀⠀⠀⠀⠀⠀

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

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

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

$:\implies\sf \sqrt{225 \times 202500}$

$:\implies\sf \sqrt{45562500}$

$:\implies\sf{\boxed{\frak{\pink{6750\;m^2}}}}\;\bigstar$

$\therefore\;{\underline{\sf{Hence,\;Area\;of\;triangle\;is\; \bf{6750}.}}}$