A) \[O{{A}^{2}}+O{{B}^{2}}\]
B) \[O{{B}^{2}}+O{{C}^{2}}\]
C) \[O{{C}^{2}}+O{{D}^{2}}\]
D) \[A{{C}^{2}}+B{{D}^{2}}\]
Correct Answer: D
Solution :
Let the length of sides of rhombus be x, length of OC be \[{{x}_{1}}\] and length of OD be \[{{y}_{1}}\]Then, \[A{{B}^{2}}+B{{C}^{2}}+C{{D}^{2}}+D{{A}^{2}}=4{{x}^{2}}\] Since, AC and DB bisect each other at O. \[\therefore \] \[AC=2{{x}_{1}}\] and \[BD=2{{y}_{1}}\] In \[\Delta AOD,\text{ }\Delta DOC,\text{ }\Delta AOB,\text{ }\Delta BOC\] \[4\left[ x_{1}^{2}+y_{1}^{2} \right]=4{{x}^{2}}\] \[\Rightarrow \] \[A{{C}^{2}}+B{{D}^{2}}=A{{B}^{2}}+B{{C}^{2}}+C{{D}^{2}}+A{{D}^{2}}\]
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