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

Explanation

Given, perimeter of a triangular field is `420 m` and its sides are in the ratio `6:7:8.`
Let sides of a triangular field be `a=6x, b=7x` and `c=8x.`
`therefore"Perimeter of a triangular field",2s=a+b+c`
`rArr" "420=6x+7x+8xrArr420=21x`
`rArr" "x=(420)/(21)=20m`
`therefore"Side of a triangular field are"`
`therefore"Side of a triangular field are"`
`a=6xx20=120m`
`b=7xx20=140m`
`"and"" "c=8xx20=160m.`
`"Now"," ""semi-perimeter", S=(a+b+c)/(2)=(120+140+160)/(2)`
`=(420)/(20)=210m`
`therefore"Area of a triangular field"=sqrt(s(s-a)(s-b)(s-c))" "["by Heron's formula"]`
`=sqrt(210(210-120)(210-140)(210-160))`
`=sqrt(210xx90xx70xx50)`
`100=sqrt(21xx9xx7xx5)`
`100=sqrt(7xx3xx3^(2)xx7xx5)`
`=100xx7xx3xxsqrt15=2100sqrt15m^(2)`
Hence, the area of triangular field is `2100sqrt15m^(2)`.