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

Question

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

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Emery 1 month 2021-08-16T06:41:16+00:00 2 Answers 0 views 0

Answers ( )

    0
    2021-08-16T06:42:24+00:00

    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 .

    ⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

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

    Exigency to find : Area of Triangle .

    ⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

    ❍ 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}}}}\\

    ⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

    ⠀⠀⠀⠀⠀ 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}}}}\\

    ⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

    0
    2021-08-16T06:42:48+00:00

    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.

    ⠀⠀━━━━━━━━━━━━━━━━━━━⠀

    « 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.

    ⠀⠀━━━━━━━━━━━━━━━━━━━⠀

    « 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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