Question

Two sides of a triangular field ase 85m and
154m in length and its perimeter is 324m
And the area of the field.​

Answer 1

Given that,

• Two sides of a triangular field are 85 m and 154 m.

We've to find out third side.

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☯ Let the third side be x m.

• Given that, Perimeter of triangular filed is 324 m.

$:\implies\sf 85 + 154 + x = 324 \\\\\\:\implies\sf 239 + x = 324 \\\\\\:\implies\sf x = 324 - 239\\\\\\:\implies{\underline{\boxed{\frak{\pink{x = 85 \ m}}}}}$

Hence, third side of the rectangular field is 85 m.

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$\qquad\quad\boxed{\bf{\mid{\overline{\underline{\pink{\bigstar\: Using \ Heron's \ Formula \ :}}}}}\mid}$

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$\bf{Sides}\begin{cases}\sf{\:\: a = 85 \ m}\\\sf{\ \ b = 154 \ m}\\\sf{\ \ c = 85 \ m}\end{cases}$

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$:\implies\sf S_{\:(Semiperimeter)} = \dfrac{a + b + c}{2} \\\\\\:\implies\sf S_{\:(Semiperimeter)} = \dfrac{85 + 154 + 85}{2} \\\\\\:\implies\sf S_{\:(Semiperimeter)} = \cancel\dfrac{342}{2} \\\\\\:\implies{\underline{\boxed{\frak{\pink{S_{\:(Semiperimeter)} = 162 \ m }}}}}$

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

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$:\implies\sf Area = \sqrt{162(162 - 85) (162 - 154) (262 - 85)} \\\\\\:\implies\sf Area = \sqrt{ 162 \times 77 \times 8 \times 77} \\\\\\:\implies\sf Area = \sqrt{ 2 \times 9 \times 9 \times 7 \times 11 \times 2 \times 2 \times 2 \times 7 \times 11} \\\\\\:\implies\sf Area = \sqrt{11 \times 9 \times 7 \times 2 \times 2} \\\\\\:\implies{\underline{\boxed{\frak{\purple{Area = 2772 \ m^2}}}}} \ \bigstar$

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$\therefore\:{\underline{\sf{Hence,\: Area \ of \ the \ triangular \ field \:is\;\bf{2772 \ m^2}.}}}$

Answer 2

$\huge\star\:\:{\orange{\underline{\pink{\mathbf{Answer}}}}}$

$\:\sf\underline\green{Question\:Given}\:$

• Two sides of a triangular field are 85m and 154m in length and its perimeter is 324m And the area of the field.

$\:\sf\underline\blue{Given\: parameters}\:$

• One side of the triangular field is 85 m
• Third side measures 154 m
• The perimeter of the field is 324 m

$\:\sf\underline\orange{To\:find}\:$

• The third side
• The area of the field

$\:\sf\underline\red{Formulas\:used}\:$

• Perimeter of triangle = Sum of all sides
• Area of a triangle = $\sf\:\sqrt{s(s\:-\:a)(s\:-\:b)(s\:-\:c)}$

• Semi perimeter = $\sf\dfrac{a\:+\:b\:+\:c}{2}$

$\:\sf\underline\pink{Required\: solution}\:$

If we assume the third side as x then we get ,

• Perimeter = Sum of all sides
• 324 = 85 + 154 + x
• 324 = 239 + x
• 324 - 239 = x
• 85 m = x

As we have found the length of the third side

Let us find the semi perimeter ,

• Semi perimeter = $\sf\dfrac{a\:+\:b\:+\:c}{2}$

• S = $\sf\dfrac{85\:+\:154\:+\:85}{2}$

• S = $\sf\dfrac{324}{2}$

• S = 162 m

Now substitute the value of s in the Heron's formula .

We get the following result ,

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

• $\sf\:\sqrt{162\:(162\:-\:85)\:(162\:-\:154)\:(162\:-\:85)}$

• $\sf\:\sqrt{162\:(77)\:(8)\:(77)}$

• Area = 2772 m²