Question 10 AB isa diameter of a circle and AC is the chord such that ∠ BAC = 30∘. If the tangent at C intersects AB extended at D, then BC = BD.
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Solution
True To prove, BC = BD
Join BC and OC Given∠BAC=30∘∠BCD=30∘ ⇒[ angle between tangent and chord is equal to angle made by chord in the alternate segment] ∴∠ACD=∠ACO+∠OCD=30∘+90∘=120∘[OC⊥CDandOA=OC=radius⇒∠OAC=∠OCA=30∘InΔACD,∠CAD+∠ACD+∠ADC=180∘ [ Since sum of all interior angles of a triangle is 180∘] ⇒30∘+120∘+∠ADC=180∘⇒∠ADC=180∘−(30∘+120∘)=30∘Now,inΔBCD∠BCD=∠BDC=30∘⇒BC=BD [Since , sides opposite to equal angles are equal]