Page 1 of 1

geometry

Posted: Wed Nov 04, 2015 8:56 pm
by Mahfuz Sobhan
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 $$∠BAO = ∠BAC$$.

Re: geometry

Posted: Sat Nov 21, 2015 8:18 pm
by Mallika Prova
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,$OB=OA$.

Re: geometry

Posted: Sun Nov 22, 2015 1:18 am
by sowmitra
$BD$ will be a tangent $\Leftrightarrow OB\perp BD \Leftrightarrow OB||AC \Leftrightarrow \angle ABO=\angle BAC \Leftrightarrow \angle BAO=\angle BAC$

Re: geometry

Posted: Thu Feb 21, 2019 11:47 pm
by samiul_samin
Mahfuz Sobhan wrote:
Wed Nov 04, 2015 8:56 pm
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 $$∠BAO = ∠BAC$$.
This is BdMO National 2014 Secondary P4 & Higher Secondary P3.