Question

Sides of a triangles are in the ratio of 12:17:25 and it's perimeter is 540cm. Find it's area using heron's formula​

Answer 1

Given, ratio of sides of triangle is 12:17:25. And, Perimeter of triangle is 540 cm.

Let's consider the sides of triangles a, b and c be 12x, 17x and 25x.

⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━⠀⠀⠀⠀

We know that, Perimeter of a triangle is the sum of its sides. Then,

⠀⠀⠀⠀

$\qquad\qquad\dashrightarrow\sf 12x + 17x + 25x = 540\\\\\\ \qquad\qquad\dashrightarrow\sf 54x = 540\\\\\\ \qquad\qquad\dashrightarrow\sf x = \cancel{\dfrac{540}{54}}\\\\\\ \qquad\qquad\dashrightarrow\sf {\underline{\boxed{\pmb{\frak{x = 10}}}}}\:\bigstar$

$\therefore$ Sides of triangle will be 120 cm, 170 cm & 250 cm.

⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━⠀⠀⠀⠀

If Perimeter of triangle is 540 cm. Then, Semi-perimeter (s) will be 270 cm.

⠀⠀⠀⠀

$\bigstar\:{\underline{\sf{Using\:Heron's\:formula\:to\:find\:Area\:of\:\triangle\::}}}$

$\bigstar\:{\underline{\boxed{\pmb{\sf{Area_{\:(triangle)} = \sqrt{s\bigg(s - a\bigg)\bigg(s - b\bigg)\bigg(s - c\bigg)}}}}}}$

$\dashrightarrow\sf \sqrt{s\bigg(270 - 120\bigg)\bigg(270 - 170\bigg)\bigg(270 - 250\bigg)} \\\\\\ \dashrightarrow\sf \sqrt{250 \times 150 \times 100 \times 20}\\\\\\ \dashrightarrow\sf {\underline{\boxed{\pmb{\frak{\pink{9000\:cm^2}}}}}}\:\bigstar$

$\therefore\:{\underline{\sf{Area\:of\:triangle\:is\:{\pmb{9000\:cm^2}}}}}.$

Answer 2

Given :-

Sides of a triangles are in the ratio of 12:17:25 and it's perimeter is 540cm.

To Find :-

Area

Solution :-

Let the sides be 12x, 17x and 25x

Perimeter = a + b + c

540 = 12x + 17x + 25x

540 = 54x

540/54 = x

10 = x

Therefore

Sides are

12x = 12(10) = 120 cm

17x = 17(10) = 170 cm

25x = 25(10) = 250 cm

Now

Semiperimeter = Perimeter/2

Semiperimeter = 540/2

Semiperimeter = 270 cm

Now

Area = √s(s - a)(s - b)(s - c)

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

Area = √270 × 150 × 100 × 20

Area = √(8,10,00,000)

Area = 9000 cm²

Therefore

Area of triangle is 9000 cm²