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

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

2 thoughts on “<br />The sides of a triangle are in the ratio 3:57<br />I and its perimeter is 600m find the area of<br />triangle​”

  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 :

    [tex]\dag\:\:\it{ As,\:We\:know\:that\::}\\[/tex]

    [tex]\qquad \dag\:\:\bigg\lgroup \sf{Perimeter _{(Triangle)} \:: a + b + c }\bigg\rgroup \\\\[/tex]

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

    ⠀⠀⠀⠀⠀⠀[tex]\underline {\boldsymbol{\star\:Now \: By \: Substituting \: the \: known \: Values \::}}\\[/tex]

    [tex]\qquad \longmapsto \sf 3x + 5x + 7x = 600 \\\\ [/tex]

    [tex]\qquad \longmapsto \sf 8x + 7x = 600 \\\\ [/tex]

    [tex]\qquad \longmapsto \sf 15x = 600 \\\\ [/tex]

    [tex]\qquad \longmapsto \sf x =\cancel {\dfrac{600}{15}} \\\\ [/tex]

    [tex]\qquad \longmapsto \frak{\underline{\purple{\:x = 40m }} }\bigstar \\[/tex]

    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,

    ⠀⠀⠀⠀⠀[tex]\therefore {\underline{ \mathrm {\:Three \:Side\:of\:Triangle \:are\:\bf{ 120m\:200m\:\&\:280m}}}}\\[/tex]

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

    [tex]\dag\:\:\it{ As,\:We\:know\:that\::}\\[/tex]

    [tex]\qquad \dag\:\:\bigg\lgroup \sf{Semi-Perimeter _{(Triangle)} \:: \dfrac{Perimeter}{2} }\bigg\rgroup \\\\[/tex]

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

    ⠀⠀⠀⠀⠀⠀[tex]\underline {\boldsymbol{\star\:Now \: By \: Substituting \: the \: known \: Values \::}}\\[/tex]

    [tex]\qquad \longmapsto \sf Semi-Perimeter \:= \dfrac{600}{2}\\[/tex]

    [tex]\qquad \longmapsto \sf Semi-Perimeter \:= \cancel {\dfrac{600}{2}}\\[/tex]

    [tex]\qquad \longmapsto \frak{\underline{\purple{\:Semi-Perimeter \:= 300 \: m }} }\bigstar \\[/tex]

    Therefore,

    ⠀⠀⠀⠀⠀[tex]\therefore {\underline{ \mathrm {\:Semi-Perimeter \:of\:Triangle \:is\:\bf{300\:m}}}}\\[/tex]

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    ⠀⠀⠀⠀⠀ Finding Area of Triangle :

    [tex]\dag\:\:\it{ As,\:We\:know\:that\::}\\[/tex]

    [tex]\qquad \dag\:\:\bigg\lgroup \sf{Area _{(Triangle)} \:: \sqrt { s (s – a) (s – b) (s – c)} }\bigg\rgroup \\\\[/tex]

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

    ⠀⠀⠀⠀⠀⠀[tex]\underline {\boldsymbol{\star\:Now \: By \: Substituting \: the \: known \: Values \::}}\\[/tex]

    [tex]\qquad \longmapsto \sf \sqrt { 300 (300 – 120) (300 – 200) (300 – 280)} \\ [/tex]

    [tex]\qquad \longmapsto \sf \sqrt { 300 (180) (300 – 200) (300 – 280)} \\ [/tex]

    [tex]\qquad \longmapsto \sf \sqrt { 300 (180) \times (100) \times (20)} \\ [/tex]

    [tex]\qquad \longmapsto \sf \sqrt { 300 \times 18,000 \times 20} \\ [/tex]

    [tex]\qquad \longmapsto \sf \sqrt { 6,000 \times 18,000 } \\ [/tex]

    [tex]\qquad \longmapsto \sf \sqrt { 10,80,00,000 } \\ [/tex]

    [tex]\qquad \longmapsto \frak{\underline{\purple{\:Area =6000 \sqrt {3} m ^2 }} }\bigstar \\[/tex]

    Therefore,

    ⠀⠀⠀⠀⠀[tex]\therefore {\underline{ \mathrm {\:Area \:of\:Triangle \:is\:\bf{ 6000 \sqrt {3} \:m^2}}}}\\[/tex]

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  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(sa)(sb)(sc)

    • 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.

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