the area of a triangle is 150cm² and its sides are in the ratio 3 : 4 : 5.what is its perimeter .
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the area of a triangle is 150cm² and its sides are in the ratio 3 : 4 : 5.what is its perimeter .
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Answer -
To find -
Step-by-step explanation -
Let -
We know that -
[tex]\underline{\boxed{\sf Area_{(triangle)} = \sqrt{s \: (s - a) \: (s - b) \: (s - c)}}}[/tex]
Where -
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We know that -
[tex]\underline{\boxed{\sf Semi-Perimeter = \dfrac{a + b +c}{2}}}[/tex]
Where -
Here -
Therefore -
[tex]\tt \dashrightarrow Semi-Perimeter = \dfrac{3x + 4x + 5x}{2}[/tex]
[tex]\tt \dashrightarrow Semi-Perimeter = \dfrac{12x}{2}[/tex]
[tex]\tt \dashrightarrow Semi-Perimeter = 6x[/tex]
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Now, applying the formula of area of a triangle -
[tex]\rm 150 = \sqrt{6x \:(6x - 3x) \: (6x - 4x) \: (6x - 5x)}[/tex]
[tex]\rm 150 = \sqrt{6x \: (3x) \: (2x) \: (x)}[/tex]
[tex]\rm 150 = \sqrt{6x \times 3x \times 2x \times x}[/tex]
[tex]\rm 150 = \sqrt{36x^4}[/tex]
[tex]\rm 150 = 6x^2[/tex]
[tex]\rm \dfrac{150}{6} = x^2[/tex]
[tex]\rm 25 = x^2[/tex]
[tex]\rm \sqrt{25} = x[/tex]
[tex]\rm 5\: cm = x[/tex]
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Hence, the sides of the triangle are :-
[tex]\bf \mapsto 3x = 3 \times 5 = 15 \: cm.[/tex]
[tex]\bf \mapsto 4x = 4 \times 5 = 20 \: cm.[/tex]
[tex]\bf \mapsto 5x = 5 \times 5 = 25 \: cm.[/tex]
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We know that -
[tex]\underline{\boxed{\sf Perimeter_{(triangle)} = Sum \: of \: all \: sides}}[/tex]
Here -
Therefore -
[tex]\bf \longmapsto Perimeter = 15 + 20 + 25[/tex]
[tex]\bf \longmapsto Perimeter = 60 \: cm[/tex]
[tex] \\ [/tex]
Hence -
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