question_answer

ABC is a right triangle with AB = AC. Bisector of \[\angle A\]meets BC at D. Then

A)  BC = 2AD      

B)         2BC = AD                   

C)         AB = BC       

D)         None of these

Correct Answer: A

Solution :

\[\Delta ABC\]is right angle triangle with AB = AC and AD is the bisector of\[\angle A.\] Now in \[\Delta ABC,AB=BC\Rightarrow \angle C=\angle B\]          ?(1) [Angles opposite to equal sides are equal] Now, in \[\Delta ABC,\angle A={{90}^{o}}\] and \[\angle A+\angle B+\angle C={{180}^{o}}\] [Angle sum property of\[\Delta \]] \[\Rightarrow \]\[{{90}^{o}}+\angle B+\angle B={{180}^{o}}\] [From (1)] \[\Rightarrow \]\[2\angle B={{90}^{o}}\Rightarrow \angle B={{45}^{o}}\] \[\Rightarrow \]\[\angle B=\angle C={{45}^{o}}\]or \[\angle 3=\angle 4={{45}^{o}}\] Also, \[\angle 1=\angle 2={{45}^{o}}\][\[\because \]AD is bisector of \[\angle A\]] Also, \[\angle 1=\angle 3,\angle 2=\angle 4={{45}^{o}}\] \[\Rightarrow \]BD = AD, DC = AD                                  ?(2) [Sides opposite to equal angles are equal] Thus, BC = BD + DC = AD + AD [From (2)] \[\Rightarrow \]\[BC=2AD\]