geometry

For students of class 9-10 (age 14-16)
Mahfuz Sobhan
Posts: 25
Joined: Sat Feb 07, 2015 5:40 pm

geometry

Unread post by Mahfuz Sobhan » 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 $$∠BAO = ∠BAC$$.

User avatar
Mallika Prova
Posts: 6
Joined: Thu Dec 05, 2013 7:44 pm
Location: Mymensingh,Bangladesh

Re: geometry

Unread post by Mallika Prova » Sat Nov 21, 2015 8:18 pm

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,$OB=OA$.

User avatar
sowmitra
Posts: 155
Joined: Tue Mar 20, 2012 12:55 am
Location: Mirpur, Dhaka, Bangladesh

Re: geometry

Unread post by sowmitra » Sun Nov 22, 2015 1:18 am

$BD$ will be a tangent $\Leftrightarrow OB\perp BD \Leftrightarrow OB||AC \Leftrightarrow \angle ABO=\angle BAC \Leftrightarrow \angle BAO=\angle BAC$
"Rhythm is mathematics of the sub-conscious."
Some-Angle Related Problems;

User avatar
samiul_samin
Posts: 999
Joined: Sat Dec 09, 2017 1:32 pm

Re: geometry

Unread post by samiul_samin » Thu Feb 21, 2019 11:47 pm

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.

Post Reply