Question
Wed March 04, 2015 By: Sanmit Ratnaparkhi

If PA and PB are tangents to a circle with centre O and OP=diameter of circle .prove that APB is an equilateral triangle.

Expert Reply
Vimala Ramamurthy
Fri March 06, 2015
C o n s i d e r space t h e space f o l l o w i n g space f i g u r e.
S i n c e space r a d i u s space i s space p e r p e n d i c u l a r space t o space t h e space tan g e n t space a t space t h e space p o i n t space o f space c o n t a c t comma space w e space h a v e comma O A perpendicular P A space a n d space O B perpendicular P B rightwards double arrow angle O A P equals angle O B P equals 90 degree C o n s i d e r space t h e space r i g h t space t r i a n g l e space O A P space a n d space l e t space theta space b e space t h e space a n g l e comma space angle O P A G v i e n space t h a t space O P equals 2 O A rightwards double arrow 1 half equals fraction numerator O A over denominator O P end fraction rightwards double arrow 1 half equals fraction numerator o p p over denominator h y p end fraction rightwards double arrow 1 half equals sin theta rightwards double arrow sin 30 degree equals sin theta rightwards double arrow theta equals 30 degree T h e space l i n e space f r o m space t h e space e x t e r n a l space p o i n t space i s space t h e space a n g l e space b i s e c t o r space b e t w e e n space t h e space tan g e n t s rightwards double arrow O P space i s space t h e space a n g l e space b i s e c t o r space i n angle A P B rightwards double arrow angle O P A equals angle O P B equals theta T h u s comma space angle A P B equals angle O P A plus angle O P B space space space space space space space space space space space space space space space space space space space space equals 30 degree plus 30 degree space space space space space space space space angle A P B space equals 60 degree W e space k n o w space t h a t space t h e space tan g e n t s space f r o m space a n space e x t e r n a l space p o i n t space t o space t h e space c i r c l e space a r e space e q u a l. T h u s comma space P A equals P B T h e r e f o r e comma space triangle P A B space i s space a n space i s o s c e l e s space t r i a n g l e. H e n c e space b a s e space a n g l e s space a r e space e q u a l. rightwards double arrow angle P A B equals angle P B A B y space a n g l e space s u m space p r o p e r t y comma space w e space h a v e comma angle A P B plus angle P A B plus angle P B A equals 180 degree rightwards double arrow angle A P B plus angle P A B plus angle P A B equals 180 degree rightwards double arrow 60 degree plus 2 angle P A B equals 180 degree rightwards double arrow 2 angle P A B equals 180 degree minus 60 degree rightwards double arrow 2 angle P A B equals 120 degree rightwards double arrow angle P A B equals fraction numerator 120 degree over denominator 2 end fraction rightwards double arrow angle P A B equals 60 degree rightwards double arrow angle P A B equals angle P B A equals 60 degree rightwards double arrow angle P A B equals angle P B A equals angle A P B equals 60 degree T h u s comma space triangle A P B space i s space a n space e q u i l a t e r a l space t r i a n g l e
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