question_answer

In a triangle ABC if \[A\equiv (1,2)\] and internal angle bisectors through B and C are \[y=x\] and \[y=-2x,\] then the in radius r of \[\Delta ABC\] is

A)  \[1\sqrt{3}\]     

B)  \[1\sqrt{2}\]

C)  2/3      

D)  None of these

Correct Answer: B

Solution :

Images of A about y \[y=x,\]\[y=-2x\] are \[{{A}_{1}}\] and \[{{A}_{2}}\] which lie on BC. Now, \[{{A}_{1}}\equiv (2,1)\] and \[{{A}_{2}}\equiv (-11/5,2/5).\] Equation of BC is \[x-7y+5=0.\] Hence, \[r=ID=\left| \frac{5}{\sqrt{1+49}} \right|=\frac{5}{5\sqrt{2}}=\frac{1}{\sqrt{2}}\]