| Consider the following statements |
| I. Let ABCD be a parallelogram which is not a rectangle. Then, \[2\,(A{{B}^{2}}+B{{C}^{2}})\ne A{{C}^{2}}+B{{D}^{2}}\] |
| II. If ABCD is a rhombus with AB = 4 cm, then \[A{{C}^{2}}+B{{D}^{2}}={{n}^{3}}\]for some positive integer n. Which of the above statements Ware Correct? |
A) Only I
B) Only II
C) Both I and II
D) Neither I nor II
Correct Answer: B
Solution :
| I. ABCD is a parallelogram, then |
| \[A{{C}^{2}}+B{{D}^{2}}=2\,(A{{B}^{2}}+B{{C}^{2}})\] |
 |
| II. ABCD is a rhombus and diagonals AC and BD bisect each other. |
| \[\therefore \] \[AO=OC\] and \[OB=OD\] |
| In \[\Delta AOB,\]\[A{{B}^{2}}=A{{O}^{2}}+O{{B}^{2}}\] |
| \[\Rightarrow \] \[{{(4)}^{2}}={{\left( \frac{AC}{2} \right)}^{2}}+{{\left( \frac{BD}{2} \right)}^{2}}\] |
| \[\therefore \] \[A{{C}^{2}}+B{{D}^{2}}=64={{(4)}^{3}}\]i.e. \[{{n}^{3}}\] |
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