question_answer

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. \[{{n}^{3}}\]