Question

Find the area of triangle, two sides of which are 8 cm and 11 cm and the perimeter is 32 cm

Answer 1

Given : Two sides of the ∆ are 8 cm and 11 cm. & the perimeter is 32 cm.

Let's consider, the third side be x cm.

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• We know that :

★ Perimeter of ∆ = (a + b + c) ★

➺ 8 + 11 + x = 32

➺ 19 + x = 32

➺ x = 32 - 19

➺ x = 13

The Third side of the triangle is 13 cm.⠀⠀⠀⠀

» As we know that semi perimeter of the triangle is sum of all sides i.e ( s ) = (a + b + c)/2. Therefore,

$</p><p>\begin{gathered}\dashrightarrow\sf s = \dfrac{a + b + c}{2}\\\\\\\dashrightarrow\sf s = \dfrac{8 + 11 + 13}{2}\\\\\\\dashrightarrow\sf s = \cancel\dfrac{32}{2}\\\\\\\dashrightarrow\underline{\boxed{\pmb{\frak{s = 16\;cm}}}}\\\\\end{gathered}$

Therefore, Semi perimeter of the given triangle is 16 cm.

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$\begin{gathered}\bigstar\:{\underline{\bf{Using\:Heron's\:formula\:to\:find\:Area\:of\:\triangle\::}}}\\\\\end{gathered} </p><p>$

$\begin{gathered}\bigstar\:{\underline{\boxed{\pmb{\sf{Area_{\: (triangle)} = \sqrt{s\bigg(s - a\bigg)\bigg(s - b\bigg)\bigg(s - c\bigg)}}}}}}\\\\\end{gathered}$

$\begin{gathered}\dashrightarrow\sf Area_{\;(triangle)} = \sqrt{16\Big(16 - 8\Big)\Big(16 - 11\Big)\Big(16 - 13\Big)}\\\\\\\dashrightarrow\sf Area_{\;(triangle)} = \sqrt{16 \times 8 \times 5 \times 3}\\\\\\\dashrightarrow\sf Area_{\;(triangle)} =8\sqrt{30}\\\\\\\dashrightarrow{\underline{\boxed{\pmb{\frak{\pink{8 \sqrt{30} \:cm^2}}}}}}\:\bigstar\\\\\\ \therefore\:{\underline{\sf{Area\:of\:the\;triangle\:is\:{\sf{\pmb{8 \sqrt{30} \:cm^2}}}}}}.\end{gathered}$