Question

The sides of a triangle are y/z+z/,z/x+x/y and x/y+y/z the its area in square units is

Answer 1

Answer:

Sqr [$x^{2}$/$y^{2}$ +  $y^{2}$/$z^{2}$ + $z^{2}$/$x^{2}$ + 2(xy/yz + xz/xy + yz/xz)]

Step-by-step explanation:

The given sides of a triangle are;

a = y/z+z/x

b = z/x+x/y

c = x/y+y/z

We can find the area of the triangle by using Heron's formula;

Area = Sqr (s x (s-a) x (s-b) x (s-c))

where a, b, c are the sides of the triangle and s = (a + b c)/2

s = (a + b c)/2

  = (y/z + z/x + z/x + x/y + x/y + y/z)/2

  = 2 x (x/y + y/z + z/x) / 2

  = x/y + y/z + z/x

Area = Sqr (s x (s-a) x (s-b) x (s-c))

        = Sqr ((x/y + y/z + z/x) x (x/y) x (y/z) x (z/x))

        = Sqr ($x^{2}$/$y^{2}$ + xy/yz + xz/xy + xy/yz + $y^{2}$/$z^{2}$ + yz/xz + xz/xy + yz/xz + $z^{2}$/$x^{2}$)

        = Sqr [$x^{2}$/$y^{2}$ +  $y^{2}$/$z^{2}$ + $z^{2}$/$x^{2}$ + 2(xy/yz + xz/xy + yz/xz)]

Answer 2

Answer:

Step-by-step explanation:

The given sides of a triangle are;

a = y/z+z/x

b = z/x+x/y

c = x/y+y/z

We can find the area of the triangle by using Heron's formula;

Area = Sqr (s x (s-a) x (s-b) x (s-c))

where a, b, c are the sides of the triangle and s = (a + b c)/2

s = (a + b c)/2

 = (y/z + z/x + z/x + x/y + x/y + y/z)/2

 = 2 x (x/y + y/z + z/x) / 2

 = x/y + y/z + z/x

Area = Sqr (s x (s-a) x (s-b) x (s-c))

       = Sqr ((x/y + y/z + z/x) x (x/y) x (y/z) x (z/x))

       = Sqr (/ + xy/yz + xz/xy + xy/yz + / + yz/xz + xz/xy + yz/xz + /)

       = Sqr [/ +  / + / + 2(xy/yz + xz/xy + yz/xz)]