A) \[A{{D}^{2}}+B{{C}^{2}}\]
B) \[A{{O}^{2}}+O{{C}^{2}}\]
C) \[A{{C}^{2}}+B{{D}^{2}}\]
D) \[2(A{{O}^{2}}+O{{B}^{2}})\]
Correct Answer: C
Solution :
As diagonals of a rhombus bisect each other at right angles. \[\Rightarrow \] \[AO=OC\]and\[BO=OD\] Applying Pythagoras theorem \[\Delta AOB\,\,\Delta AOD,\,\,\Delta DOC,\,\,\Delta BOC\]and on adding \[A{{B}^{2}}+B{{C}^{2}}+C{{D}^{2}}+A{{D}^{2}}\] \[=2[A{{O}^{2}}+O{{C}^{2}}+B{{O}^{2}}+D{{O}^{2}}]\] \[=2[2A{{O}^{2}}+2B{{O}^{2}}]\] \[=4[A{{O}^{2}}+B{{O}^{2}}]\] \[\left[ \because \,\,AO=\frac{AC}{2},\,\,BO=\frac{BD}{2} \right]\] \[=A{{C}^{2}}+B{{D}^{2}}\] fYou need to login to perform this action.
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