Question

The area of a triangle whose sides are 13 cm, 14cm and 15 cm is....
(a) 84 sq.cm
(b) 64 sq.cm.
(c) 825 sq.cm
(d) none of these​

Answer 1

☀️ Given that, sides of a triangle are 13cm, 14cm, 15cm and we need to find the area of that triangle.

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To calculate the area of the triangle we have to solve this by the Heron's Formula ::

$\boxed{\large \underline{\bf \bigstar \;\; A = \sqrt{s(s-a)(s-b)(s-c)}}}$

Where,

• A is area of triangle
• s is semi-perimeter
• a,b and c are sides

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T H E R E F O R E,

• Semi-Perimeter = Perimeter/2

⤳  Semi-Perimeter = a+b+c/2

⤳  Semi-Perimeter = (13+14+15)/2

⤳  Semi-Perimeter = 42/2

⤳  Semi-Perimeter = 21cm

Finding the area :-

⤳  √21 × ( 21 - 13 ) × ( 21 - 14 ) × ( 21 - 15 )

⤳  √21 × 8 × 7 × 6

⤳  √168 × 42

⤳  √7056

⤳  84 cm²

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• Henceforth, area of the triangle is 84cm² [ Option 'a' ]

Answer 2

Answer:

$\begin{gathered}{\large{\textsf{\textbf{\underline{\underline{Given :}}}}}}\end{gathered}$

• ➳ Sides of a Triangle ️ = 13 cm, 14 cm, and 15 cm

$\begin{gathered}\end{gathered}$

$\begin{gathered}{\large{\textsf{\textbf{\underline{\underline{To Find :}}}}}}\end{gathered}$

• ➳ Area of Triangle ️

$\begin{gathered}\end{gathered}$

$\begin{gathered}{\large{\textsf{\textbf{\underline{\underline{Using Formulae :}}}}}}\end{gathered}$

$\dag{\underline{\boxed{\sf{Semi \: Perimeter = \dfrac{a + b + c}{2}}}}}$

$\dag{\underline{\boxed{\sf{A = \sqrt{s(s - a)(s - b)(s - c)}}}}}$

$\begin{gathered}\end{gathered}$

$\begin{gathered}{\large{\textsf{\textbf{\underline{\underline{Solution :}}}}}}\end{gathered}$

$\dag{\underline{\underline{\pmb{\frak{\red{Here}}}}}}$

$\quad : \rightarrow{\sf{a = 13 cm}}$

$\quad : \rightarrow{\sf{b = 14 cm}}$

$\quad : \rightarrow{\sf{c = 15 cm}}$

$\begin{gathered}\end{gathered}$

$\dag{\underline{\underline{\pmb{\frak{\red{Finding \: the \: area \: of \: Triangle \: by \: heron's \: formula}}}}}}$

$\red\bigstar$ Calculating the semi perimeter

$\quad{: \implies{\sf{Semi \: Perimeter = \dfrac{a + b + c}{2}}}}$

• Substuting the values

$\quad{: \implies{\sf{Semi \: Perimeter = \dfrac{13 + 14 + 15}{2}}}}$

$\quad{: \implies{\sf{Semi \: Perimeter = \dfrac{42}{2}}}}$

$\quad{: \implies{\sf{Semi \: Perimeter = \cancel{\dfrac{42}{2}}}}}$

$\quad{: \implies{\sf{Semi \: Perimeter = 21 \: cm}}}$

$\begin{gathered}\end{gathered}$

$\red\bigstar$ Now, area of Triangle

$\quad{: \longmapsto{\sf{Area \: of \: \triangle = \sqrt{s(s - a)(s - b)(s - c)}}}}$

• Substuting the values

$\quad{: \longmapsto{\sf{Area \: of \: \triangle = \sqrt{21(21- 13)(21- 14)(21- 15)}}}}$

$\quad{: \longmapsto{\sf{Area \: of \: \triangle = \sqrt{21(8)(7)(6)}}}}$

$\quad{: \longmapsto{\sf{Area \: of \: \triangle = \sqrt{21 \times 8 \times 7 \times 6}}}}$

$\quad{: \longmapsto{\sf{Area \: of \: \triangle = \sqrt{7056}}}}$

$\quad{: \longmapsto{\sf{Area \: of \: \triangle = \sqrt{84 \times 84}}}}$

$\quad{: \longmapsto{\sf{Area \: of \: \triangle = 84 \: {cm}^{2} }}}$

$\bigstar\underline{\boxed{\sf{\blue{Area \: of \: Triangle} = \purple{84 \: {cm}^{2}}}}}$

∴ The option A) 84 cm² is the correct answer.

$\begin{gathered}\end{gathered}$

$\begin{gathered}{\large{\textsf{\textbf{\underline{\underline{Learn More :}}}}}}\end{gathered}$

${: \implies{\sf{Area \: of \: Triangle = \dfrac{1}{2} \bigg( b \times h\bigg)}}}$

${ : \implies{\sf{Perimeter \: of \: Triangle = a + b + c}}}$

Where

• ➳ B = Base
• ➳ H = Height
• ➳ a,b and c = Sides of Triangle

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