Question

sides of a triangle are in the ratio of 12:17:25 and it's perimeter is 540 cm. find it's area​

Answer 1

SOLUTION

Ratio of the sides of the triangle = 12 : 17 : 25Let the common ratio be x then sides are 12x, 17x and 25xPerimeter of the triangle = 540cm12x + 17x + 25x = 540 cm⇒ 54x = 540cm⇒ x = 10Sides of triangle are,12x = 12 × 10 = 120cm17x = 17 × 10 = 170cm25x = 25 × 10 = 250cmSemi perimeter of triangle(s) = 540/2 = 270cmUsing heron's formula,Area of the triangle = √s (s-a) (s-b) (s-c)                                       = √270(270 - 120) (270 - 170) (270 - 250)cm2                                       = √270 × 150 × 100 × 20 cm2                                       = 9000 cm2

Answer 2

Sides of ∆ are in ratio 12:17:25.

Perimeter of triangle is 540 cm.

⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━━━

☯ Let sides of triangle be 12x, 17x and 25x.

We know that,

Perimeter of a triangle = Sum of measure of all its sides.

$:\implies\sf 12x + 17x + 25x = 540$

$\qquad\quad:\implies\sf 54x = 540$

$\qquad\quad:\implies\sf x = { \dfrac{540}{54}}$

$\qquad\quad:\implies{\boxed{\sf{\pink{x = 10}}}}\;\bigstar$

Therefore, Sides of ∆ will be,

• 12x = 12 × 10 = 120 cm

• 17x = 17 × 10 = 170 cm

• 25x = 25 × 10 = 250 cm

⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━━━

⠀⠀⠀⠀⠀ Now, Using Heron's Formula,

$\star\;{\boxed{\sf{\purple{ \sqrt{s(s - a)(s - b)(s - c)}}}}}$

Where,

$\qquad\quad\sf s = \dfrac{a + b + c}{2}$

$\dashrightarrow\sf s = \dfrac{120 + 170 + 250}{2}$

$\qquad\quad\dashrightarrow\sf s = { \dfrac{540}{2}}$

$\quad\quad\dashrightarrow\bf s = 270\;cm$

Putting values in formula,

$:\implies\sf \sqrt{270(270 - 120)(270 - 170)(270 - 250)}$

$\qquad \qquad:\implies\sf \sqrt{270 \times 150 \times 100 \times 20}$

$\qquad\qquad\qquad:\implies{\boxed{\sf{\pink{ 9000\;cm^2}}}}\;\bigstar$

$\therefore$ Area of given triangle is 9000 cm².