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Question
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.
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Solution
Given: The perimeter of a triangular field is 420 m and its sides are in the ratio 6 : 7 : 8.
According to the question, Let the sides in meters are a = 6x, b = 7x and c = 8x.
So, perimeter of the triangle = 6x + 7x + 8x
420 = 21x
x = `420/21`
x = 20
Since, the sides of the triangular field are a = 6 × 20 cm = 120 m, b = 7 × 20 m = 140 m and c = 8 × 20 m = 160 m.
Now, semi-perimeter(s) of triangle will be:
`s = 1/2 xx 420 m`
= 210 m
Area of the triangle field = `sqrt(s(s - a)(s - b)(s - c))` ...[Using Heron’s formula]
= `sqrt(210(210 - 120)(210 - 140)(210 - 160))`
= `sqrt(210 xx 90 xx 70 xx 50)`
= `100sqrt(7 xx 3 xx 3^2 xx 7 xx 5)`
= `100 xx 7 xx 3 xx sqrt(15)`
= `2100sqrt(15)`
Therefore, the area of the triangular field is `2100sqrt(15)`.
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