Question

5. Find the area of a triangular field whose sides are 91 m, 98 m and 105 m
in length. Find the height corresponding to the longest side.
​

Answer 1

$\huge\mathcal\pink{Question:-}$

Find the area of a triangular field whose sides are 91 m, 98 m and 105 m in length. Find the height corresponding to the longest side.

$\huge\mathcal\pink{Answer:-}$

Δ= s(s−a)(s−b)(s−c)

Δ= √147×56×49×42

Δ=4116m ^2

Base=105m

Height =h

h=78.4m.

Answer 2

Given: Sides of triangular field are 91 m, 98 m and 105 m.

To find: The height corresponding to the longest side?

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

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

$:\implies\sf s = semi - perimeter$

$:\implies\sf s = \dfrac{a + b + c}{2}$

$:\implies\sf s = \dfrac{91 + 98 + 105}{2}$

$:\implies\sf s = \cancel{ \dfrac{294}{2}}$

$:\implies{\underline{\boxed{\frak{\purple{s = 147\:m}}}}}\;\bigstar$

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$\bf{\dag}\;{\underline{\frak{Now,\:Putting\:values\:in\:formula,}}}$

$:\implies\sf Area_{\;(field)} = \sqrt{147(147 - 91)(147 - 98)(147 - 105)}$

$:\implies\sf Area_{\;(field)} = \sqrt{147 \times 56 \times 49 \times 42}$

$:\implies\sf Area_{\;(field)} = \sqrt{16941456}$

$:\implies{\underline{\boxed{\frak{\purple{Area_{\;(field)} = 4116\:m^2}}}}}\;\bigstar$

$\therefore\:{\underline{\sf{Area\:of\:triangular\:field\:is\: \bf{4116\:m^2}.}}}$

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☯ Let's Consider h as the height corresponding to the longest side.

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Here,

• 105 m is the longest side = Base

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$\dag\;{\underline{\frak{As\;we\;know\;that,}}}$

$\star\;{\boxed{\sf{\pink{Area_{\;(triangle)} = \dfrac{1}{2} \times base \times height}}}}$

$:\implies\sf \dfrac{1}{2} \times 105 \times h = 4116$

$:\implies\sf 105 \times h = 4116 \times 2$

$:\implies\sf 105 \times h = 8232$

$:\implies\sf h = \cancel{ \dfrac{8232}{105}}$

$:\implies{\underline{\boxed{\frak{\purple{Area_{\;(field)} = 78.4\:m}}}}}\;\bigstar$

$\therefore\:{\underline{\sf{Hence,\:the\:height\: corresponding\:to\:the\;longest\: side\:is\; {\textsf{\textbf{78.4\:m}}}.}}}$