​solve q number 20 step by step
Q.20. The perimeter of a rhombus is 240 cm and one of its diagonals is 80 cm. Find its area using Herons formula.

Dear Student,

$$\begin{gathered} As, \\ Perimeter of rhombus=240 cm \\ side of rhombus=\frac{240}{4}=60cm \left[All sides of rhombus are equal\right] \\ Diagonal=80 cm \\ Now, \\ Area of triangle u\sin g Heron\text{'}s formula=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)} \\ Where s=\frac{sum of sides of triangle}{2}=\frac{AB+BC+BD}{2} \left(in ∆ABD\right) \\ s=\frac{60+60+80}{2}=\frac{200}{2}=100 \\ So, \\ Area =\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)} \\ where s=100 , a=60, b=60 ,c=80 \\ Area=\sqrt{100\left(100-60\right)\left(100-60\right)\left(100-80\right)} \\ =\sqrt{100\times 40\times 40\times 20} \\ =10\times 40\sqrt{2\times 2\times 5} \\ =10\times 40\times 2\sqrt{5} \\ =800\sqrt{5} \\ Now, \\ Ar\left(∆ABD\right)=Ar\left(∆CDB\right)=800\sqrt{5} \\ Ar\left(ABCD\right)=Ar\left(∆ABD\right)+Ar\left(∆CDB\right) \\ =800\sqrt{5}+800\sqrt{5} \\ =\boxed{\mathbf{1600}\sqrt{\mathbf{5}}} \end{gathered}$$

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