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

Correct Question:-

The perimeter of a triangle is 450m . The ratio of its side's are 13:12:5. Using Heron's formula find the area of the triangle.

Answer:-

$\red{\bigstar}$ Area of the triangle$\large\leadsto\boxed{\tt\purple{6750 \: m^2}}$

• Given:-

• Perimeter of the triangle = 450 m

• Ratio of their sides = 13:12:5

• To Find:-

• Area of the triangle.

• Solution:-

Let the sides of the triangle in ratio be 13x , 12x and 5x.

Given that,

Perimeter of the triangle is 450 m.

Hence,

➪ $\sf 450 = 13x + 12x + 5x$

➪ $\sf 30x = 450$

➪ $\sf x = \dfrac{450}{30}$

★ $\bf\pink{x = 15}$

Therefore,

• 13x = 13 × 15 = 195 m

• 12x = 12 × 15 = 180 m

• 5x = 5 × 15 = 75 m

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We know, by Heron's formula:-

$\underline{\boxed{\bf\red{Area \: of \: triangle = \sqrt{s(s-a)(s-b)(s-c)}}}}$

Firstly, finding the semi - perimeter (s):-

➪ $\sf s = \dfrac{a + b + c}{2}$

➪ $\sf s = \dfrac{195 + 180 + 75}{2}$

➪ $\sf s = \dfrac{450}{2}$

★ $\bf\pink{s = 225 \: m}$

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• Substituting in the Heron's Formula:-

$\sf Area = \sqrt{225 (225-195)(225-180)(225-75)}$

➪ $\sf Area = \sqrt{225 \times 30 \times 45 \times 150}$

➪ $\sf Area = \sqrt{225 \times 1350 \times 150}$

➪ $\sf Area = \sqrt{303750 \times 150}$

➪ $\sf Area = \sqrt{45562500}$

★ $\large{\bf\pink{Area = 6750 \: m^2}}$

Therefore, the area of the triangle is 6750 m².