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Write ‘True’ or ‘False’ and justify your answer in each of the following
AB is a diameter of a circle and AC is its chord such that \angleBAC = 30°. If the tangent at C intersects AB extended at D, then BC = BD.

Answers (1)

Answer True
Solution: First of all we solve the question according to give conditions. If we able to prove it then it will be true otherwise it will be false.
Given :\angleBAC = 30^{\circ}
Diagram : Construct figure according to given conditions then join BC and OC.

To Prove : BC = BD
Proof :\angleBAC = 30^{\circ}    (Given)
\Rightarrow \, \angle BCD= 30^{\circ}
[\because angle between chord and tangent id equal to the angle made by chord in alternate segment]
\angle OCD= 90^{\circ} 
[\because Radius and tangent’s angle is always 90^{\circ}]
In \bigtriangleupOAC                    
OA = OC     (both are radius of circle)
\angle OCA= 30^{\circ}
\Rightarrow \angle OCA= 30^{\circ}  [opposite angles of an isosceles triangle is equal]
\therefore \; \; \angle ACD= \angle ACO+\angle OCD
= 30+90= 120^{\circ}
In \bigtriangleup ACD
\angle CAD+\angle ADC+\angle DCA= 180^{\circ}
    [\because sum of interior angle of a trianglE 180^{\circ}]
30^{\circ}+\angle ADC+120^{\circ}= 180^{\circ}
\angle ADC= 180^{\circ}-120^{\circ}-30^{\circ}
< ADC=30^{\circ}
In \bigtriangleupBCD we conclude that
  \angle BCD= 30^{\circ} and\angle ADC=30^{\circ} 
\Rightarrow BC= BD[\becausesides which is opposite to equal angles is always equal]
Hence Proved.
Hence the given statement is true.


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