question_answer

One diagonal of a rhombus is half the other. If the length of the side of the rhombus is 10 cm, what is the area of the rhombus?

A) \[75\sqrt{3}\,sq.\,cm.\]               

B) \[25\sqrt{3}\,sq.\,cm.\]

C) \[80\,sq.\,cm.\]              

D) \[160\,sq.\,cm.\]    

Correct Answer: C

Solution :

The diagonals of a rhombus-bisect each other at right angles.             \[\therefore \]\[\angle AOB={{90}^{o}}:AO=OC:OD=OB\]             If \[AC=2x\,cm.\,BD=x\,cm.\]             In \[\Delta \Alpha {\mathrm O}\Beta \] \[A{{B}^{2}}=O{{A}^{2}}+O{{B}^{2}}\]             \[\Rightarrow \]\[{{10}^{2}}={{x}^{2}}+{{\left( \frac{x}{2} \right)}^{2}}\]             \[\Rightarrow \]\[{{x}^{2}}+\frac{{{x}^{2}}}{4}=100\]           \[\Rightarrow \]\[\frac{5{{x}^{2}}4}{4}=100\] \[\Rightarrow \]\[{{x}^{2}}=\frac{100\times 4}{5}=80\]              ? (i) \[\therefore \]Area of rhombus \[=\frac{1}{2}\times {{d}_{1}}\times {{d}_{2}}\] \[=\frac{1}{2}\times 2x\times x\] \[={{x}^{2}}=80\,sq.\,cm.\]