Question

The sides of a triangular field are in the ratio 5:3:4 and its perimeter is 180m. Find its area.

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Answer 1

Step-by-step explanation:

Given:-

• The sides of a triangular field = 5 : 3 : 4

• The perimeter of the triangle = 180cm.

To Find:-

• The area of the triangle

Solution:-

Let the common ratio between the sides be x

So, the sides of the triangle are 5x , 3x and 4x

The perimeter of the triangle = Sum of its sides.

$\sf \implies 5x + 3x + 4x = 180$

$\sf \implies 12x = 180$

$\sf \implies x = \dfrac{180}{12}$

$\sf \implies x = 15$

1st side (a) = 5x = 5 × 15 = 75cm

2nd side (b) = 3x = 3 × 15 = 45cm

3rd side (c) = 4x = 4 × 15 = 60cm

Semi- perimeter (s) = $\sf \dfrac{Perimeter}{2} = \dfrac{180}{2} = 90$

$\sf Area\: of \: triangle = \sqrt{s(s - a)(s-b)(s-c)}$

$\sf = \sqrt{90(90 - 75)(90-45)(90-60)}$

$\sf = \sqrt{90(90-45)(90-60)}$

$\sf = \sqrt{90 \times 15 \times 45 \times 30}$

$\sf = \sqrt{1822500}$

$\sf = 1350m^2$

$\underline{\boxed{\sf \therefore Area \: of \: the \: triangle = 1350m^2}}$

Answer 2

Given :-

Ratio of the sides of the triangle = 5 : 3 : 4

Perimeter of the triangle = 180 m

To Find :-

The area of the triangle.

Analysis :-

Consider the common ratio as a variable.

Make an equation accordingly such that the three sides of the triangle is multiplied to the variable each added is equal to the perimeter of the triangle.

Find the value of the variable and substitute them in the sides and find them accordingly.

Next, we should find the semi perimeter by dividing the perimeter given by two.

Finally, substitute the values we got using Heron's formula and find the area accordingly.

Solution :-

We know that,

• p = Perimeter
• a = Area
• s = Semi perimeter

Let the common ratio be 'x'. Then the sides would be 5x, 3x and 4x.

Given that,

Perimeter (p) = 180 m

Making an equation,

5x + 3x + 4x = 180

12x = 180

By transposing,

x = 180/12

x = 15

Finding the sides,

5x = 5 × 15 = 75 m

3x = 3 × 15 = 45 m

4x = 4 × 15 = 60 m

Therefore, the sides of the triangle are 75 m, 45 m and 60 m.

Finding the semi perimeter,

$\underline{\boxed{\sf Semi \ perimeter=\dfrac{Perimeter}{2} }}$

Given that,

Perimeter (p) = 180 m

Substituting their values,

Semi perimeter = 180/2

Semi perimeter = 90 m

Therefore, the semi perimeter of the triangle is 90 m.

Using Heron's formula,

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

Given that,

Semi perimeter (p) = 90 m

First side = 75 m

Second side = 45 m

Third side = 60 m

Substituting their values,

$\sf =\sqrt{90(90-75)(90-45)(90-60)}$

$\sf =\sqrt{90 \times 15 \times 45 \times 30}$

$\sf =\sqrt{ 1822500 }$

$\sf =1350 \ m^2$

Therefore, the area of the triangle is 1350 m².