question_answer

In an equilateral \[\Delta ABC,\] if \[AD\bot BC,\] then which of the following is true?

A) \[2A{{B}^{2}}=3A{{D}^{2}}\]         

B) \[4A{{B}^{2}}=3A{{D}^{2}}\]

C) \[3\,\,A{{B}^{2}}=4A{{D}^{2}}\]

D) \[3A{{B}^{2}}=2A{{D}^{2}}\]

Correct Answer: C

Solution :

\[AB=BC=CA\]

\[AD\bot BC\]\[\Rightarrow \]\[BD=DC\]

In \[\Delta ABD,\]\[A{{B}^{2}}=B{{D}^{2}}+A{{D}^{2}}\]

\[\Rightarrow \]\[A{{B}^{2}}={{\left( \frac{1}{2}AB \right)}^{2}}+A{{D}^{2}}\]

\[\Rightarrow \]\[A{{B}^{2}}-\frac{1}{4}A{{B}^{2}}=A{{D}^{2}}\]\[\Rightarrow \]\[3A{{B}^{2}}=4A{{D}^{2}}\]