question_answer

In \[\Delta ABC,\]\[AD\bot BC,\] then 

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

B)             \[A{{B}^{2}}+B{{D}^{2}}=A{{C}^{2}}-C{{D}^{2}}\]  

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

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

Correct Answer: A

Solution :

  In \[\Delta ABD,\] \[A{{B}^{2}}=A{{D}^{2}}+B{{D}^{2}}\]               ...(i) In right angled \[\Delta ACD,\] we have \[A{{C}^{2}}=A{{D}^{2}}+C{{D}^{2}}\]               ...(ii) On subtracting Eq. (ii) from Eq. (i), we get \[A{{B}^{2}}-A{{C}^{2}}=B{{D}^{2}}-C{{D}^{2}}\] \[A{{B}^{2}}-B{{D}^{2}}=A{{C}^{2}}-C{{D}^{2}}\]