Question

$|sides \: of \: a \: triangle \: are \: in \: the \: ratio \: of \: 12 : 17 : 25 \: and \: its \: perimeter \: is \: 540cm \: find \: the \: aera \:$
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Answer 1

Answer:Let the sides triangle be 12x ,17x and 25x

perimeter of triangle=540cm

12x+17x+25x=540cm

54x=540cm

x=10cm

Thus,triangle's sides are 120cm,170cm,250cm

Area of triangle by heron's formula

then

semi perimeter =120+170+250/2

=540/2=270cm

Area

=^270 *(270-120)*(270-170)*(270-250)

=^270*150*100*20

=^3*3*3*5*2*5*3*5*2*5*2*5*2*5*2*2

=3*3*5*5*5*2*2*2

=9000cm^2

so, ar ea of triangle is 9000cm^2

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Answer 2

❒ Let the sides of the triangle be 12x, 17x and 25x.

Given that,

• Perimeter of the triangle is 540 cm.

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Therefore,

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$:\implies\sf 12x + 17x + 25x = 540 \\\\\\:\implies\sf 29x + 25x = 540 \\\\\\:\implies\sf 54x = 540\\\\\\:\implies\sf x = \cancel\dfrac{540}{54}\\\\\\:\implies{\underline{\boxed{\frak{\pink{x = 10}}}}}\:\bigstar$

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❒ Hence, all Sides of triangle are:

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• 12x = 12(10) = 120 cm
• 17x = 17(10) = 170 cm
• 25x = 25(10) = 250 cm

$\therefore\:{\underline{\sf{Hence, \ sides \ of \ the \ triangle \ are \: \bf{120 m, \ 170m \ 250m}.}}}$

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$\underline{\bf{\dag} \:\mathfrak{Finding\: Semi- perimeter\: :}}$

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$\dag\:\boxed{\sf{\pink{S_{\:(semi - perimeter)} = \dfrac{a + b + c}{2}}}} \\\\\\:\implies\sf s = \dfrac{120 + 170 + 250}{2} \\\\\\:\implies\sf s = \cancel\dfrac{ 540}{2}\\\\\\:\implies{\underline{\boxed{\frak{\purple{ s = 270 \: cm}}}}}\:\bigstar$ ⠀⠀⠀⠀

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$\therefore\:{\underline{\sf{Hence, \ semi \ perimeter \ of \ the \ \triangle \ is\: \bf{270 \ cm}.}}}$⠀

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$\underline{\bf{\dag} \:\mathfrak{As \: we \; know \: that\: :}}$

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$\dag\:\boxed{\sf{\pink{Area_{\triangle} = \sqrt{s(s - a)(s - b) (s - c)}}}} \\\\\\:\implies\sf Area_{\triangle} = \sqrt{270(270 - 120) (270 - 170) (270 - 250)} \\\\\\:\implies\sf Area_{\triangle} = \sqrt{270 \times 150 \times 100 \times 20} \\\\\\:\implies{\underline{\boxed{\frak{\purple{Area_{\triangle} = 9000\: cm^2}}}}}\:\bigstar$ ⠀⠀⠀⠀⠀

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$\therefore\:{\underline{\sf{Hence, \ area \ of \ the \ \triangle \ is\: \bf{9000\: cm^2}.}}}$