Question

the perimeter of a triangle in 540m and its sides are in the ratio 25:17:12 find the area of triangle by herons formula​

Answer 1

Step-by-step explanation:

GIVEN : The perimeter of a triangular field = 540m

Let the sides are 25x , 17x , 12 x

Perimeter of a ∆ = sum of three sides

25x + 17x + 12x = 540

54x = 540

x = 10

1st side (a) - 25x = 25×10= 250m

2nd side(b)= 17x = 17×10= 170m

3rd side (c)= 12x = 12 × 10 =120m

Semi - perimeter ( S) = a+b+c/2

= (250 + 170+120)/2 = 540/2 = 270 m

Area of the ∆= √ S(S - a)(S - b)(S - c)

[By Heron’s Formula]

= √ S(S - 250)(S - 170)(S - 120)

= √ 270(270 - 250)(270 - 170)(270 - 120)

= √ 270× 20×100×150

= √ 81000000

Area of the ∆= 9000 m²

Hence, the Area of the ∆= 9000 m²

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Answer 2

Answer: 9000m²

Step-by-step explanation:

Ratio of the sides = 25:17:12

Let the length of the sides be 25x , 17x  , 12x

Perimeter = 540 m .

∴ 25x + 17x + 12x = 540

⇒ 54x = 540

⇒ x = 10.

∴ The lengths of the sides = 250m,170m,120m

Heron's Formula = $\sqrt{s(s-a)(s-b)(s-c)}$ (where s is the semi-perimeter)

Here, s = 540/2 = 270 m , a = 250 m , b = 170 m , c = 120 m .

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

⇒ $\sqrt{270(270 - 250)(270 - 170)(270 - 120)}$

⇒ $\sqrt{270 * 20 * 100 * 150}$

⇒ $\sqrt{2 * 3^3 * 5 * 2^2 * 5 * 2^2 * 5^2 * 2 * 3 * 5^2}$

⇒ $\sqrt{2^6 * 3^4 * 5^6}$

⇒ $2^3 * 3^2 * 5^3$

⇒ 8 * 9 * 125 = 9000m²

∴ Area of the triangle = 9000m²