question_answer

Let s denote the semi perimeter of a triangle ABC in which BC = a, CA = b, AB = c. If a circle touches the sides BC, CA, AB at D, E, F, respectively, find BD.

A) \[s-b\]              

B)        \[2s+h\]                      

C)        \[b+s\]            

D)        \[3b-s\]

Correct Answer: A

Solution :

According to question, \[s=\frac{a+b+c}{2}\,\,\Rightarrow \,\,2s=a+b+c\]             B is external point and BD and BF are tangents and from an external point the   tangents drawn to a circle are equal in length.     \[\therefore \]    \[BD=BF;~\] \[AF=AE;\] \[CD=CE\] s = Semi perimeter \[=\frac{AB+AC+BC}{2}\] \[2s=AB+AC+BC\] \[2s=AF+FB+AE+EC+BD+DC\] \[\Rightarrow \]            \[2s=2AE+2CE+2BD\] \[\Rightarrow \]            \[s=AE+CE+BD\] \[\Rightarrow \] \[s=AC+BD\Rightarrow s-b=BD\].