Question

the sides of triangle are in the ratio 25 : 17 : 12 the and its perimeter is 540 m. find the sides of triangle and its area ?​​

Answer 1

✧ Given :

• Triangle are in the ratio 25 : 17 : 12
• It's perimeter is 540 m

✧ To Find :

• Sides of triangle and It's area ?

✧ Solution :

We know that,

Perimeter of ∆ = Sum of all sides

Perimeter of ∆ = a + b + c

➙ 540 = 25x + 17x + 12x

➙ 540 = 54x

➙ 540/54 = x

➙ 10 = x

➙ A = 25x = 25 × 10 = 250 m

➙ B = 17x = 17 × 10 = 170 m

➙ C = 12x = 12 × 10 = 120 m

■ Hence the sides of triangle is 250 m, 170 m, 120 m.

We know that,

➙ Semi - perimeter = a + b + c / 2

Now, substitute the values :-

➙ Semi - perimeter = 250 + 170 + 120 / 2

➙ Semi - perimeter = 540 / 2

➙ Semi - perimeter = 270 m

We also know that,

Area of ∆ = √S (s-a) (s-b) (s-c)

Now substitute the values :-

➙ Area of ∆ = √270 (270-250) (270-170) (270-120)

➙ Area of ∆ = √√270 × 20 × 100 × 150

➙ Area of ∆ = √270 × 2000 × 150

➙ Area of ∆ = √270 × 300000

➙ Area of ∆ = √81000000

➙ Area of ∆ = 9000 m²

■ Hence, the area of triangle is 9000 m².

Answer 2

$\begin{gathered}\begin{gathered}\bf \: Given \: - \begin{cases} &\sf{sides \: are \: in \: the \: ratio \: 25 : 17 : 12} \\ &\sf{perimeter \: = \: 540 \: m} \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf \: To \: Find \: - \begin{cases} &\sf{sides \: of \: triangle} \\ &\sf{area \: of \: triangle}  \end{cases}\end{gathered}\end{gathered}$

$\large\underline\purple{\bold{Solution :- }}$

Let the sides of a triangle be a, b and c respectively.

Since a : b : c = 25 : 17 : 12

$\begin{gathered}\begin{gathered}\bf \: So -   \begin{cases} &\sf{a \: = \: 25 \: x} \\ &\sf{b \: = \: 17 \: x} \\ &\sf{c \: = \: 12 \: x} \end{cases}\end{gathered}\end{gathered}$

$\underline{\boxed{ \pink{{\rm \: Perimeter \ of \ a \ triangle=(a+b+c)}}}}$

Now, Perimeter = 540 m

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

$\rm : \implies \:54x \: = \: 540$

$\boxed{ \pink{\rm : \implies \:x = 10}}$

$\begin{gathered}\begin{gathered}\bf \: So, \: sides \: of \: \triangle \: are -   \begin{cases} &\sf{a \: = \: 25 \times 10 = 250 \: m} \\ &\sf{b \: = \: 17 \times 10 = 170 \: m} \\ &\sf{c \: = \: 12 \times 10 = 120 \: m} \end{cases}\end{gathered}\end{gathered}$

Now, to find the area of triangle

$\begin{gathered}\begin{gathered}\bf \: Sides \: are \: -   \begin{cases} &\sf{a \: = \: 250 \: m} \\ &\sf{b \: = \: 170 \: m} \\ &\sf{c \: = \: 120 \: m} \end{cases}\end{gathered}\end{gathered}$

$\underline{\boxed{ \pink{\sf Semi \: Perimeter \ of \ a \ triangle, \: s \: = \dfrac{1}{2} (a+b+c)}}}$

$\rm : \implies \:s \: = \: \dfrac{250 + 170 + 120}{2}$

$\rm : \implies \:s \: = \: \dfrac{540}{2}$

$\rm : \implies \:s \: = \: 270 \: m$

Hence,

Area of triangle is given by

$\underline{\boxed { \green{\rm Area \ of \ triangle=\sqrt{s(s-a)(s-b)(s-c)} }}}$

$\rm : \implies \:Area \: = \: \sqrt{270(270 - 250)(270 - 150)(270 - 120)}$

$\rm : \implies \:Area \: = \sqrt{270 \times 20 \times 100 \times 150}$

$\boxed{ \pink{\rm : \implies \:Area \: = 9000 \: {m}^{2} }}$