Question

Perimeter of triangular field is 135cm sides in a ratio of 25:17:12 find area using herons formula

Answer 1

Step-by-step explanation:

The perimeter of a triangular field is 240 m, and its sides are in the ratio 25:17:12. Find the area of the field. Also, find the cost of ploughing the field at Rs.40 per m2

Answer 2

$\red{\bigstar} G I V E N$

• The perimeter of the triangular field is 135 cm
• The ratio of sides of triangular field is 25:17:12

$\blue{\bigstar} T OㅤF I N D$

• Area of the triangular field ?

$\purple{\bigstar} S O L U T I O N$

• Let sides of the triangular field be 25x, 17x, 12x

$\quad\odot\:\underline{\boxed{\sf{Perimeter_{(triangular\:field)} = Sum\:of\:all\:sides\:of\:triangular\:field}}}$

Putting all known values ::

$\sf \quad \dashrightarrow\quad 135 = 25x + 17x + 12x$

$\begin{gathered}\\ \sf \quad \dashrightarrow\quad 54x = 135\end{gathered}$

$\begin{gathered}\\ \sf \quad \dashrightarrow\quad x = {\cancel{\dfrac{135}{54}}}\end{gathered}$

$\begin{gathered}\\ \sf \quad \dashrightarrow\quad \pink{x = 2.5\:cm}\end{gathered}$

$\green{\bigstar} H E N C E$

Sides of triangular field ::

• 1st side = 25x = 25 × 2.5 = 62.5 cm
• 2nd side = 17x = 17 × 2.5 = 42.5 cm
• 3rd side = 12x = 12 × 2.5 = 30 cm

We know that ::

$\quad\odot\:\underline{\boxed{\sf{Semi\:Perimeter_{(triangular\:field)}\:(s) = \dfrac{Perimeter_{(triangular\:field)}}{2}}}}$

Putting all known values ::

$\sf \quad \dashrightarrow\quad Semi\:Perimeter_{(triangular\:field)}\:(s) = \dfrac{135}{2}$

After cancelling 135 with 2, we get ::

$\sf \quad \dashrightarrow\quad\pink{Semi\:Perimeter_{(triangular\:field)}\:(s) = 67.5\:cm}$

Now, finding area of triangular field ::

Using heron's formula ::

$\quad\odot\:\underline{\boxed{\sf{Area_{(triangular\:field)} = \sqrt{s\:(s - a)\:(s - b)\:(s - c)}}}}$

Where a, b, and c are sides of triangular field.

Putting all known values ::

$\sf \quad \dashrightarrow\quad Area_{(triangular\:field)} = \sqrt{67.5\:(67.5 - 62.5)\:(67.5 - 42.5)\:(67.5 - 30)}$

$\begin{gathered}\\ \sf \quad \dashrightarrow\quad Area_{(triangular\:field)} = \sqrt{67.5\:\times\:5\:\times\:25\:\times\:37.5}\end{gathered}$

$\begin{gathered}\\ \sf \quad \dashrightarrow\quad Area_{(triangular\:field)} = \sqrt{316406.25}\end{gathered}$

$\begin{gathered}\\ \sf \quad \dashrightarrow\quad \pink{Area_{(triangular\:field)} = 562.5\:cm^2}\end{gathered}$

$\therefore\:{\underline{\sf{Hence,\:area\:of\:triangular\:field\:is\:\bf{562.5\:cm^2}}}}$