Question

A triangle shaped land has the sides in the ratio 3:5:7, if its perimeter is 300 meter. What will be the area of the land ?​

Answer 1

Given :-

Ratio of the sides of a triangle shaped land = 3 : 4 : 7

Perimeter of the triangle shaped land = 300 m

To Find :-

Sides of the triangle.

The area of the land.

Solution :-

We know that,

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

Given that,

Ratio = 3 : 4 : 7

Perimeter (p) = 300 m

Let the sides of the triangle be 3x, 5x and 7x respectively.

According to the question,

$\sf 3x + 5x + 7x=300$

$\sf 15x=300$

$\sf x=\dfrac{300}{15}$

$\sf x=20$

Length of side a = $\sf 3x= 3 \times 20=60 \ m$

Length of side b = $\sf 5x= 5 \times 20=100 \ m$

Length of side c = $\sf 7x = 7 \times 20=140 \ m$

Given that, perimeter = 300 m

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

Substituting their values, we get

Semi-perimeter = $\sf \dfrac{60+100+140}{2} =150 \ m$

Therefore, the semi-perimeter is 150 m

Using Heron's formula,

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

Substituting their values,

= $\sf \sqrt{150(150-60)(150-100)(150-140)}$

= $\sf \sqrt{150 \times 90 \times 60 \times 10}$

= $\sf \sqrt{30\times5 \times 30 \times 3 \times 5\times 10\times 10 }$

$\sf =30 \times 5 \times 10\sqrt{3}$

$\sf =1500\sqrt{3}$

Therefore, the area of the land is 1500√3 cm²