Question

the area of a triangle is 150cm² and its sides are in the ratio 3 : 4 : 5.what is its perimeter .​

Answer 1

Answer -

• The perimeter of the triangle = 60 cm.

To find -

• The perimeter of the triangle.

Step-by-step explanation -

• Here, it is given that the area of a triangle is 150 cm² and it's sides are in the ratio 3 : 4 : 5. We have to find it's perimeter.

Let -

• The sides be 3x, 4x and 5x.

We know that -

$\underline{\boxed{\sf Area_{(triangle)} = \sqrt{s \: (s - a) \: (s - b) \: (s - c)}}}$

Where -

• s = Semi-perimeter.
• a = First side.
• b = Second side.
• c = Third side.

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• Let's find the semi-perimeter of the triangle first!

We know that -

$\underline{\boxed{\sf Semi-Perimeter = \dfrac{a + b +c}{2}}}$

Where -

• a = First side.
• b = Second side.
• c = Third side.

Here -

• First side = 3x.
• Second side = 4x.
• Third side = 5x.

Therefore -

$\tt \dashrightarrow Semi-Perimeter = \dfrac{3x + 4x + 5x}{2}$

$\tt \dashrightarrow Semi-Perimeter = \dfrac{12x}{2}$

$\tt \dashrightarrow Semi-Perimeter = 6x$

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Now, applying the formula of area of a triangle -

$\rm 150 = \sqrt{6x \:(6x - 3x) \: (6x - 4x) \: (6x - 5x)}$

$\rm 150 = \sqrt{6x \: (3x) \: (2x) \: (x)}$

$\rm 150 = \sqrt{6x \times 3x \times 2x \times x}$

$\rm 150 = \sqrt{36x^4}$

$\rm 150 = 6x^2$

$\rm \dfrac{150}{6} = x^2$

$\rm 25 = x^2$

$\rm \sqrt{25} = x$

$\rm 5\: cm = x$

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Hence, the sides of the triangle are :-

$\bf \mapsto 3x = 3 \times 5 = 15 \: cm.$

$\bf \mapsto 4x = 4 \times 5 = 20 \: cm.$

$\bf \mapsto 5x = 5 \times 5 = 25 \: cm.$

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• Finally, let's find the perimeter of the triangle!

We know that -

$\underline{\boxed{\sf Perimeter_{(triangle)} = Sum \: of \: all \: sides}}$

Here -

• The sides are 15 cm, 20 cm and 25 cm.

Therefore -

$\bf \longmapsto Perimeter = 15 + 20 + 25$

$\bf \longmapsto Perimeter = 60 \: cm$

Hence -

• The perimeter of the triangle is 60 cm.

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