geometry

 Posts: 25
 Joined: Sat Feb 07, 2015 5:40 pm
geometry
In $$ΔABC$$, $$∠B = 90$$. A circle is drawn taking $$AB$$ as a chord. $$O$$ is the center of the circle. $$O$$ and $$C$$ isn't on the same side of $$AB$$. $$BD$$ is perpendicular to $$AC$$. Prove that, $$BD$$ will be a tangent to the circle if and only if $$∠BAO = ∠BAC$$.
 Mallika Prova
 Posts: 6
 Joined: Thu Dec 05, 2013 7:44 pm
 Location: Mymensingh,Bangladesh
Re: geometry
its enough to prove that $\angle BAO=\angle BAC$ when $BD$ is a tangent to the circle...
now,if $BD$ is a tangent $\angle OBD=\angle OBA+\angle ABD=90$.
again,$\angle D=90.\angle BAD+\angle ABD=90$.
then,$\angle OBA=\angle CAB$ and $\angle BAO=\angle BAC$ as,$OB=OA$.
now,if $BD$ is a tangent $\angle OBD=\angle OBA+\angle ABD=90$.
again,$\angle D=90.\angle BAD+\angle ABD=90$.
then,$\angle OBA=\angle CAB$ and $\angle BAO=\angle BAC$ as,$OB=OA$.
Re: geometry
$BD$ will be a tangent $\Leftrightarrow OB\perp BD \Leftrightarrow OBAC \Leftrightarrow \angle ABO=\angle BAC \Leftrightarrow \angle BAO=\angle BAC$
 samiul_samin
 Posts: 1004
 Joined: Sat Dec 09, 2017 1:32 pm
Re: geometry
This is BdMO National 2014 Secondary P4 & Higher Secondary P3.Mahfuz Sobhan wrote: ↑Wed Nov 04, 2015 8:56 pmIn $$ΔABC$$, $$∠B = 90$$. A circle is drawn taking $$AB$$ as a chord. $$O$$ is the center of the circle. $$O$$ and $$C$$ isn't on the same side of $$AB$$. $$BD$$ is perpendicular to $$AC$$. Prove that, $$BD$$ will be a tangent to the circle if and only if $$∠BAO = ∠BAC$$.