Question

State true or false

AB is a diameter of a circle and AC is its chord such that $\angle BAC={30}^0$ If the tangent at C intersects AB extended at D, then BC = BD.

• A. True
• B. False

Answer 1

The correct option is A True

$\Rightarrow$ $\angle ACB={90}^o$ [ Angle inscribed in a semi-circle ]

In $△ACB,$

$\angle BAC+\angle ACB+\angle CBA={180}^o$

$\Rightarrow$ ${30}^o+{90}^o+\angle CBA={180}^o$

$\Rightarrow$ ${120}^o+\angle CBA={180}^o$

$\Rightarrow$ $\angle CBA={60}^o$ -------- ( 1 )

In $△OCB,$

$OC=OB$ [ Radii of a circle ]

$\therefore$ $\angle OCB=\angle OBC$ [ Base angles of an equal sides are also equal ]

$\Rightarrow$ $\angle OCB={60}^o$

Now,

$\Rightarrow$ $\angle OCD={90}^o$ [ $OC\perp CD$ ]

$\Rightarrow$ $\angle OCB+\angle BCD={90}^o$

$\Rightarrow$ $\angle BCD={30}^o$ ------ ( 2 )

$\Rightarrow$ $\angle CBO=\angle BCD+\angle CDB$ [ Exterior angle theorem ]

$\Rightarrow$ ${60}^o={30}^o+\angle CDB$

$\Rightarrow$ $\angle CBD={30}^o$ ----- ( 3 )

From ( 2 ) and ( 3 ),

$\Rightarrow$ $\angle BCD=\angle CDB$

$\therefore$ $BC=CD$