Question

Choose the correct answer of the following question:

One side of a rhombus is 20 cm long and one of its diagonals measures 24 cm. The area of the rhombus is

(a) 192 cm² (b) 480 cm² (c) 240 cm² (d) 384 cm²

Answer 1

$$\begin{gathered} \mathrm{WE}\mathrm{have}, \\ \mathrm{AB}=\mathrm{BC}=\mathrm{CD}=\mathrm{DA}=20\mathrm{cm}\mathrm{and}\mathrm{BD}=24\mathrm{cm} \\ \mathrm{Also},\mathrm{BO}=\frac{\mathrm{BD}}{2}=\frac{24}{2}=12\mathrm{cm} \\ \mathrm{In}∆\mathrm{AOB}, \\ \mathrm{Using}\mathrm{Pythagoras}\mathrm{theorem} \\ {\mathrm{AO}}^2={\mathrm{AB}}^2-{\mathrm{BO}}^2 \\ ={20}^2-{12}^2 \\ =400-144 \\ \Rightarrow {\mathrm{AO}}^2=256 \\ \Rightarrow \mathrm{AO}=\sqrt{256} \\ \Rightarrow \mathrm{AO}=16\mathrm{cm} \\ \Rightarrow \mathrm{AC}=2\mathrm{AO}=2\times 16=32\mathrm{cm} \\ \mathrm{Now},\mathrm{the}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{rhombus}\mathrm{ABCD}=\frac{1}{2}\times \mathrm{AC}\times \mathrm{BD} \\ =\frac{1}{2}\times 32\times 24 \\ =384{\mathrm{cm}}^2 \end{gathered}$$

Hence, the correct answer is option (d).