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”
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Appropriate Question :
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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,
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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Given:
To find:
Solution:
• Let’s consider ABC is a triangle.
Where,
• 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)
→ √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.