If sides of a rhombus are 6 cm each and one diagonal is 8 cm, find the height of the rhombus.

Answer :

We form our rhombus ABCD , As  :

Now we find Area of triangle ABC by Heron's formula

We know Heron's formula :

Area of triangle  = $\sqrt{s\left(s -a\right)\left(s -b\right)\left(s -c\right)}$

Here s  = $\frac{a + b + c}{2}$ and a  =   6  b  = 6  and  c = 8  

So ,

s  = $\frac{6 + 6 + 8}{2} = \frac{20}{2} = \mathbf{10}\mathbf{ }\mathit{c}\mathit{m}$
Then
Area of triangle ABC  = $\sqrt{10\left(10 -6\right)\left(10 -6\right)\left(10 -8\right)} = \sqrt{10 \times 4 \times 4 \times 2} = \sqrt{320} = \mathbf{17}\mathbf{.}\mathbf{88}\mathbf{ }\mathit{c}{\mathit{m}}^{\mathbf{2}}$

We know diagonal of rhombus divide in to two parts , So

Area of rhombus ABCD  =  2 ( Area of triangle ABC ) = 2 ( 17.88 ) = 35.76 cm²

And

We know Area of rhombus = Height $\times$ side , So

Height of rhombus $\times$ 6 = 35.76

Height of rhombus = 5.96 cm                                           (  Ans )