Question

The sides of a triangle are in the ratio 3:57
I and its perimeter is 600m find the area of
triangle​

Answer 1

Appropriate Question :

• The Sides of Triangle are in the ratio 3:5:7 and it's Perimeter is 600 m . Find the Area of Triangle .

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Given : The Sides of Triangle are in the ratio 3:5:7 & Perimeter of Triangle is 600 m .

Exigency to find : Area of Triangle .

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❍ Let's Consider the three sides of Triangle be 3x , 5x & 7x .

⠀⠀⠀⠀⠀Finding Three Sides of Triangle :

$\dag\:\:\it{ As,\:We\:know\:that\::}$

$\qquad \dag\:\:\bigg\lgroup \sf{Perimeter _{(Triangle)} \:: a + b + c }\bigg\rgroup$

⠀⠀⠀⠀⠀Here a , b & c are three sides of Triangle & we know that Perimeter of Triangle is 400 m .

⠀⠀⠀⠀⠀⠀$\underline {\boldsymbol{\star\:Now \: By \: Substituting \: the \: known \: Values \::}}$

$\qquad \longmapsto \sf 3x + 5x + 7x = 600$

$\qquad \longmapsto \sf 8x + 7x = 600$

$\qquad \longmapsto \sf 15x = 600$

$\qquad \longmapsto \sf x =\cancel {\dfrac{600}{15}}$

$\qquad \longmapsto \frak{\underline{\purple{\:x = 40m }} }\bigstar$

Therefore,

• a or First Side is 3x = 3(40) = 120 m
• b or Second Side is 5x = 5(40) = 200 m
• c or Third side is 7x = 7(40) = 280 m

Therefore,

⠀⠀⠀⠀⠀$\therefore {\underline{ \mathrm {\:Three \:Side\:of\:Triangle \:are\:\bf{ 120m\:200m\:\&\:280m}}}}$

⠀⠀⠀⠀⠀ Finding Semi-Perimeter of Triangle for Finding Area of Triangle :

$\dag\:\:\it{ As,\:We\:know\:that\::}$

$\qquad \dag\:\:\bigg\lgroup \sf{Semi-Perimeter _{(Triangle)} \:: \dfrac{Perimeter}{2} }\bigg\rgroup$

⠀⠀⠀⠀⠀Here Perimeter of Triangle is 600 m .

⠀⠀⠀⠀⠀⠀$\underline {\boldsymbol{\star\:Now \: By \: Substituting \: the \: known \: Values \::}}$

$\qquad \longmapsto \sf Semi-Perimeter \:= \dfrac{600}{2}$

$\qquad \longmapsto \sf Semi-Perimeter \:= \cancel {\dfrac{600}{2}}$

$\qquad \longmapsto \frak{\underline{\purple{\:Semi-Perimeter \:= 300 \: m }} }\bigstar$

Therefore,

⠀⠀⠀⠀⠀$\therefore {\underline{ \mathrm {\:Semi-Perimeter \:of\:Triangle \:is\:\bf{300\:m}}}}$

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⠀⠀⠀⠀⠀ Finding Area of Triangle :

$\dag\:\:\it{ As,\:We\:know\:that\::}$

$\qquad \dag\:\:\bigg\lgroup \sf{Area _{(Triangle)} \:: \sqrt { s (s - a) (s - b) (s - c)} }\bigg\rgroup$

⠀⠀⠀⠀⠀Here a , b & c are three sides of Triangle & s is the Semi-Perimeter of Triangle.

⠀⠀⠀⠀⠀⠀$\underline {\boldsymbol{\star\:Now \: By \: Substituting \: the \: known \: Values \::}}$

$\qquad \longmapsto \sf \sqrt { 300 (300 - 120) (300 - 200) (300 - 280)}$

$\qquad \longmapsto \sf \sqrt { 300 (180) (300 - 200) (300 - 280)}$

$\qquad \longmapsto \sf \sqrt { 300 (180) \times (100) \times (20)}$

$\qquad \longmapsto \sf \sqrt { 300 \times 18,000 \times 20}$

$\qquad \longmapsto \sf \sqrt { 6,000 \times 18,000 }$

$\qquad \longmapsto \sf \sqrt { 10,80,00,000 }$

$\qquad \longmapsto \frak{\underline{\purple{\:Area =6000 \sqrt {3} m ^2 }} }\bigstar$

Therefore,

⠀⠀⠀⠀⠀$\therefore {\underline{ \mathrm {\:Area \:of\:Triangle \:is\:\bf{ 6000 \sqrt {3} \:m^2}}}}$

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Answer 2

Given:

• The sides of a triangle are in the ratio 3:5:7
• Perimeter of the triangle is 600m.

To find:

• Area of triangle?

Solution:

• Let's consider ABC is a triangle.

Where,

• A = 3x
• B = 5x
• C = 7x

• Let angle in common be x.

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« Now, By using perimeter of triangle formula,

→ Perimeter of triangle = a + b + c

→ 3x + 5x + 7x = 600

→ 15x = 600

→ x = 600/15

→ x = 40

Thus, The sides of the triangle are 120m,200m & 280m.

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« Now, Let's Find Area of triangle by using herons formula,

As we know that,

→ √s(s - a)(s - b)(s - c)

• Side = 120 + 200 + 280/2 = 300

→ √300(300 - 120)(300 - 200)(300 - 280)

→ √300(180)(100)(20)

→ √300(360000)

→ √108000000

→ 6000√3

∴ Hence, Area of of the triangle is 6000√3.