Question

Question :

The semi perimeter of a triangular ground is 450units and its sides are in the ratio 3:5:4. using heron formula find the area of the ground.​

Answer 1

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Answer 2

Solution : ↴

$\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\sf\: Assumesion \begin{cases} &\sf{\sf Let \: the \: first \: side \: be \: 3x \: units.} \\ &\sf{{\sf Second \: be \: 5x \: units. }}\\ &\sf{{\sf Third \: be \: 4x \: units. }} \end{cases}\end{gathered}\end{gathered}\end{gathered}\end{gathered}$

$\sf \: The \: perimeter \: of \: triangular \: ground = 2 \times semi-perimeter$

$\sf \implies \: 2 \times 450$

$\sf \implies \: 900 \: units$

$\sf \implies \:3x + 5x + 4x = 900$

$\sf \implies \: 12x = 900$

$\sf \implies \: x = \dfrac{900}{12}$

$\sf \implies \: x = 75$

$\underline{ \sf \: So, \: side \: of \: the \: triangular \: park \: are }:$

$\sf \: First \: side : 3x = 3 \times 75 = 225 \: units$

$\sf \: \sf \: Second\: side : 5x = 5 \times 75 = 365 \: units$

$\sf \: \sf \: Third \: side : 4x = 4 \times 75 = 300 \: units$

$\sf \: Semi-perimeter = 450 \: units \: \: \: \: \: [Already \: given]$

$\sf \: Therefore, area \: of \: triangular \: park \: is,$

$\sf \implies \: \sqrt{S(S -First \: side ) (S - Second \: side)(S - Third \: side)}$

$\sf \: Putting \: the \: values,$

$\sf \implies \: \sqrt{450(450-225 ) (450 - 375)(450 - 300)}$

$\sf \implies \: \sqrt{450 \times 225 \times 75 \times 150}$

$\sf \implies \: \sqrt{2 \times 225 \times 225 \times 75 \times 2 \times 75}$

$\sf \implies \: 2 \times 75 \times 225 = \: 33750 \: sq.units$

$\boxed{ \pink{\sf \: So, \: area \: of \: the \: ground \: is \: 33750 \: sq.units}}$