Question

The perimeter of a triangular field is 540 m and its sides are in the ratio 25:17:12

Find the area of the triangle.​

Answer 1

Given :

• The perimeter of a triangle field is 540 m.

• Sides are in the ratio 25:17:12

To find out:

Find the area of a triangle.

Formula used:

• Perimeter of triangle = AB + BC + CA

• Herons formula (Area of triangle ) = √s(s-a) (s-b) (s-c)

Solution:

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

✪According to question:-

∴Perimeter = 25x + 17x + 12x

➞ 540 = 54x

➞ x = 540/54

➞ x = 10 m

Sides of triangle:

• AB = 25x = 25 × 10 = 250 m

• BC = 17x = 17 × 10 = 170 m

• AC = 12x = 12 × 10 = 120 m

✯Semi perimeter = 540/2 = 270 m

By Herons Formula, Area of triangle field

$= \sqrt{s(s - a)(s - b)(s - c)}$

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

$= \sqrt{270 \times 20 \times 100 \times 150}$

$= \sqrt{(3 \times 3 \times3 \times10 )( 10 \times 2 )(3 \times 5 \times 10)(10 \times 10 \times 10)}$

$= \sqrt{ {3}^{4} \times {10}^{5} \times 2 \times 5} \: {m}^{2}$

$= \sqrt{ {3}^{4} \times {10}^{6} } \: {m}^{2}$

$= \sqrt{( {3}^{2} \times {10}^{3}) {}^{2} } \: {m}^{2}$

$= 9 \times 1000 \: {m}^{2}$

$= \bold{ 9000 \: m {}^{2} }$

Answer 2

QuEsTiOn :-

The perimeter of a triangular field is 540 m and its sides are in the ratio 25:17:12. find the area of the triangle .

GiVeN :-

Perimeter of triangular field = 540m

And sides are in ratio 25 : 17 : 12 .

Heron's Formula

${\purple{\boxed{\large{\bold{Area \: of \: \triangle \: = \sqrt{s(s - a)(s - b)(s - c)} }}}}}$

where a , b ,c are the sides of triangle and s is the semi perimeter.

SoLuTiOn :-

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

where , x is any positive number .

We know that ,

Perimeter of ∆ = sum of all three sides

25x + 17x + 12x = 540

54x = 540

x = 540/54

x = 10

Then ,

First side (a) = 25x = 25×10 = 250m

Second side (b) = 17x = 17×10 = 170m

Third side (c) = 12x = 12×10 = 120m

$Semi \: perimeter \: (s) = \frac{a + b + c}{2} \\ s = \frac{540}{2} \\ \\ s = 270$

So,

$Area \: of \: \triangle = \sqrt{s(s - a)(s - b)(s - c)} \\ \\ \: \: \: \: = \sqrt{270(270 - 250)(270 - 170)(270 - 120)} \\ \\ \: \: \: \: = \sqrt{270 \times 20 \times 100 \times 150} \\ \\ \: \: \: \: = \sqrt{81000000} \\ \\ \: \: \: \: = \sqrt{9000 \times 9000} \\ \\ \: \: \: \: = 9000 \: {m}^{2}$

Therefore,

${\red{\boxed{\large{\bold{The \: area \: of \: triangle \: = 9000\:sq.m }}}}}$