Question

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

Answer 1

Given:-

$\implies \sf Sides\: of \: triangle\: in \: ratio = 12:17:25 \\ \implies \sf Perimeter = 540$

To find:-

$\implies \sf Area \: of \: triangle$

Solution:-

Let, the common ratio between the sides of the given triangle be x.

So, the sides of triangle will be

12x , 17x , and 25x.

Perimeter of triangle = 540cm.

$\implies \sf 12x + 17x + 25x = 540cm \\ \\ \implies \sf 54x = 540cm \\ \\ \implies \sf x = \dfrac{540}{54} \\ \\ \implies \sf x = 10cm$

From, here we got x so now putting the value in we will get the sides of triangle .

$\implies \sf 12x = 12(10) = 120cm \\ \implies \sf 17x = 17(10) = 170cm \\ \implies \sf 25x = 25(10) = 250cm$

Now, we know that

$\large {\underline {\boxed {\red {\sf {S = \dfrac{perimeter \: of \: triangle}{2} }}}}}$

Putting the value in above formula we will get.

$\implies \sf S = \dfrac{Perimeter\: of \: triangle}{2} \\ \\ \implies \sf \dfrac{540}{2} \\ \\ \implies \sf = 270cm$

By heron's formula we will get area of triangle.

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

Therefore,the area of triangle is ${\red {\sf 9000cm^{2}}}$.