Question

The perimeter of a triangular field is 420m and its sides are in the ratio 6:7:8. Find the area of

triangular field​

Answer 1

Step 1- Given that the perimeter of the triangle is 420m.

The ratio of three sides is 6:7:8

Let's consider the lengths of the three sides are $6x$, $7x$ and $8x$.

Step 2- Perimeter of the triangle=$6x+7x+8x$

$\implies 6x+7x+8x=420\\\implies 21x=420\\\implies x=\frac{420}{21}\\\implies x=20$

• Thus lengths of the sides of triangles can be calculated by putting the value of x.
• Hence side of triangles are: 120m, 140m and 160m.

Step 3- To calculate the area of the triangle sides and semi-perimeter is required.

Semi-perimeter of the triangle,$s=\frac{perimeter}{2}=$$\frac{420}{2}=210m$

Step3- According to Heroes formula, Area$=\sqrt{s(s-a)(s-b)(s-c)}$

Where $a$, $b$ and $c$ are sides of the triangle.

Step 4- Now put the values in Heroes formula and calculate the area:

Area  $=\sqrt{210(210-120)(210-140)(210-160)} \\=\sqrt{210\cdot90\cdot70\cdot50} \\=\sqrt{21\cdot9\cdot7\cdot5\cdot10000} \\=100\sqrt{7\cdot3\cdot3\cdot3\cdot7\cdot5} \\=21\cdot100\sqrt{15}\\=2100\sqrt15 m^2$