The sides of a TRIANGLE are in ratio of 12:17:15 & the perimeter of the triangle is 540 cms. What is the area of the triangle?

Solution:

Start with simplest ASSUMPTION with the SIDE being a=12x cm, b=17x cm, 15 x cm; (as sides are in the ration 12:17:15)

Perimeter of triangle is given as 540 cm.

So, 12x + 17 x+ 15 x = 540

44 x = 540

x= 540/44 (=270/22 = 135/11)

Sides of triangle are : a=(12 * 135/11) cm; b= (17 * 135/11) cm ; c= (15 * 135/11) cm

Note: In the BEGINNING you should always quickly check that it doesn’t belong to equilateral(1:1:1) neither forms a Pythagorean triplet so not a RIGHT angle triangle.

Now, since we know all the sides so we use the below formula to find the area of triangle.

Area=SQRT(s(s-a)(s-b)(s-c))

where s=(a+b+c)/2 or perimeter/2

S= 540/2 cm

S= 270 cm (=135*2)

Area=SQRT(270(270- 12*135/11)(270–17*135/11)(270–15*135/11))

Area =SQRT(135*2(135*2- 12*135/11)(135*2–17*135/11)(135*2–15*135/11))

Area =SQRT[ 135*2{(22- 12)/11}{(22–17)/11}{(22–15)/11} ]

Area = (135)^2 * SQRT[ 2*{(22- 12)/11}{(22–17)/11}{(22–15)/11} ]

Area = (135)^2 * SQRT[ 2*(10/11)*(5/11) *(7/11) ]

Area = (135)^2 * (10/11) * SQRT[ (7/11) ]

Area = (135)^2 * (10/11) * SQRT[ (7/11) ]

Area = (135)^2 * (10/11) * 0.797

Area = 13216.83 square cm