Question

area of a triangle is 150 cm square and its sides are given in the ratio 3:4:5. what is it's perimeter? ​

Answer 1

Step-by-step explanation:

Area of triangle = 150 cm² (Given)

Area of triangle (Heron's Formula) = $\sqrt{s(s-a)(s-b)(s-c)}$

s = Semi Perimeter

Let the sides be 3x , 4x and 5x respectively.

Hence ,

a = 3x , b = 4x and c = 5x

Perimeter = Sum of all side = 3x + 4x + 5x = 12x

Semi Perimeter (s) = Perimeter/2

Therefore, Semi Perimeter = 6x

Now substitute all the values in the Heron's formula.

$\sqrt{6x(6x-3x)(6x-4x)(6x-5x)} =150cm^{2} \\\sqrt{6x*3x*2x*x} =150\\\sqrt{36x^{4} } =150\\\sqrt{[(6x)^{2}]^{2} } =150\\6x^{2} =150\\x^{2} =150/6\\x^{2} =25\\x=\sqrt{25} =5cm$

∴ x = 5cm

Perimeter = 12x = 12×5 = 60 cm

We can also verify this by adding up the sides:

Sides of the triangle =

3x = 3×5 = 15cm

4x = 4×5 = 20cm

5x = 5×5 = 25 cm

Now add up the sides:

15+20+25 = 60cm

Hence, Perimeter = 60 cm

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Answer 2

Question:

area of a triangle is 150 cm square and its sides are given in the ratio 3:4:5. what is it's perimeter? ​

To find:

what is it's perimeter? ​

Given:

area of a triangle is 150 $cm^{2}$

its sides are given in the ratio 3:4:5

Thus,

Here the ratio's are all different.Therefore we can tell that it is a            scalene triangle

The S of a scalene triangle=(Side A + Side B + Side C)/2

The area of an scalene triangle=

$\sqrt{s * (s-a) * (s-b) * (s-c)}$

Let,The 3 sides of the scalene triangle be 3x cm,4x cm,5x cm

So,S of the scalene triangle=

(3x + 4x + 5x)/2

=12x/2

=6x

So, $\sqrt{6x(6x - 3x)(6x-4x)(6x-5x)}$

    =$\sqrt{6x * 3x * 2x * x}$

     =$\sqrt{36x^{4} }$

   

    =$6x^{2}$

Therefore,

          $6x^{2}$=$150^{2}$

          or,$x^{2}$=25

          or,x=$\sqrt{5 * 5}$

          or,x=5

So,

1st side=(5 * 3)cm=15cm

2nd side=(5 * 4)cm=20cm

3rs side=(5 * 5)cm=25cm

The perimeter of scalene triangle=(Side A + Side B + Side C)

So,The perimeter of the tiangle=(15 + 20 +25)cm

                                                      =60cm

Answer:

The perimeter of the triangle=60cm