Question

The sides of a triangular park are in the ratio 12:17:25 and its perimeter is 1080m .What is its area?

Answer 1

Answer:

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Answer 2

Given: Ratio of sides of a triangular park is 12:17:25 & Perimeter of park is 1080 m.

To find: Area of triangular park?

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☯ Let the sides of triangular park a, b and c be 12x, 17x and 25x respectively.

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

$\star\;{\boxed{\sf{\pink{Perimeter_{\;(triangle)} = Sum\:of\:\:it's\:all\:sides}}}}$

$:\implies\sf a + b + c = 1080\\ \\ \\ :\implies\sf 12x + 17x + 25x = 1080\\ \\ \\ :\implies\sf 54x = 1080\\ \\ \\ :\implies\sf x = \cancel{ \dfrac{1080}{54}}\\ \\ \\ :\implies{\underline{\boxed{\frak{\purple{x = 20}}}}}\;\bigstar$

Therefore, Sides of triangle are,

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• a = 12 × 20 = 240 m
• b = 17 × 20 = 340 m
• c = 25 × 20 = 500 m

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$\underline{\bigstar\:\boldsymbol{Using\:Herons\:Formula\::}}$

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

Where,

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

$:\implies\sf s = \dfrac{a + b + c}{2}\\ \\ \\ :\implies\sf s = \dfrac{240 + 340 + 500}{2}\\ \\ \\ :\implies\sf s = \cancel{\dfrac{1080}{2}}\\ \\ \\ :\implies{\underline{\boxed{\frak{\purple{s = 540\:m}}}}}\;\bigstar$

Now,

$\sf We\:have \begin{cases} & \sf{a = \bf{240\:m}} \\ & \sf{b = \bf{340\:m}} \\ & \sf{c = \bf{500\:m}} \\ & \sf{s = \bf{540\:m}} \end{cases}$

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

$:\implies\sf \sqrt{540(540 - 240)(540 - 340)(540 - 500)}\\ \\ \\ :\implies\sf \sqrt{540 \times 300 \times 200 \times 40}$

$:\implies\sf \sqrt{(2)^2 \times (3)^2 \times 3 \times 5 \times 3 \times (10)^2 \times 2 \times (10)^2 \times (2)^2 \times 10}$

$:\implies\sf 2 \times 3 \times 10 \times 10 \times 2\sqrt{3 \times 5 \times 3 \times 2 \times 10}\\ \\ \\ :\implies\sf 1200 \sqrt{(3)^2 \times (10)^2} \\ \\ \\ :\implies\sf 1200 \times 3 \times 10\\ \\ \\ :\implies{\underline{\boxed{\frak{\purple{36000\:m^2}}}}}\;\bigstar$

$\therefore\:{\underline{\sf{Area\:of\:triangular\:park\:is\: \bf{36000\:m^2}.}}}$