## geometry

For students of class 9-10 (age 14-16)
Mahfuz Sobhan
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

### 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\$.
then,\$\angle OBA=\angle CAB\$ and \$\angle BAO=\angle BAC\$ as,\$OB=OA\$.

sowmitra
Posts: 155
Joined: Tue Mar 20, 2012 12:55 am

### Re: geometry

\$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;

samiul_samin
Posts: 1007
Joined: Sat Dec 09, 2017 1:32 pm

### Re: geometry

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\$\$.
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