question_answer

In \[\Delta ABC\], \[AD\bot BC\]and \[A{{D}^{2}}=BD\,.\,CD\]then the value of \[\angle BAC\] is

A) \[45{}^\circ \]

B) \[90{}^\circ \]

C) \[180{}^\circ \]

D) \[60{}^\circ \]

Correct Answer: B

Solution :

Given, \[\Delta ABC\] in which \[AD\bot \,BC\] and             \[A{{D}^{2}}=BD\,.\,CD\].

Now, in \[\Delta DBA\] and \[\Delta DAC\],

we have  \[\angle BDA=\angle ADC=90{}^\circ\]

\[\frac{BD}{AD}=\frac{AD}{CD}\]

\[\Delta BDA=\Delta ADC=90{}^\circ \] [by SAS similarity]

\[\angle B=\angle 2\] and \[\angle 1=\angle C\]

\[\angle 1+\angle 2=\angle B+\angle C\]

\[\angle A=\angle B+\angle C\]

\[2\angle A=\angle A+\angle B+\angle C=180{}^\circ\]

\[\angle A=\frac{180{}^\circ }{2}=90{}^\circ\]