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begin mathsize 24px style space A B thin space i s space a space d i a m e t e r space o f space t h e space c i r c l e space w i t h space C e n t r e space O. B C space a n d space A C space a r e space C h o r d s. s e g space O D thin space i s space perpendicular space C h o r d space B C. space P r o v e space t h a t space C A equals 2 D O end style

Asked by Bhawna 20th March 2015, 7:22 PM
Answered by Expert
Answer:
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.
W E space k n o w space t h a t space a n g l e space i n space a space s e m i space c i r c l e space i s space 90 degree G i v e n space t h a t space A B space i s space t h e space d i a m e t e r space o f space t h e space c i r c l e space w i t h space c e n t r e space O. T h u s space C space i s space a space p o i n t space l i e s space b e t w e e n space A space a n d space B space a n d space h e n c e space C space i s space i n space t h e space s e m i c i r c l e space A C B. T h e r e f o r e comma space angle A C B space i s space 90 degree. N o w space c o n s i d e r space t h e space t r i a n g l e space triangle B A C. triangle B A C space i s space a space r i g h t space a n g l e d space t r i a n g l e comma space r i g h t space a n g l e d space a t space C. G i v e n space t h a t space O space i s space t h e space c e n t r e space o f space t h e space c i r c l e space a n d space h e n c e space O A equals O B equals O D. A l s o space g i v e n space t h a t space O D perpendicular B C rightwards double arrow angle O D B equals 90 degree T h u s space t h e space c o r r e s p o n d i n g space a n g l e s comma space angle A C B space a n d space angle O D B space a r e space e q u a l. T h e r e f o r e comma space i t space i s space c l e a r space t h a t comma space A C space a n d space O D space a r e space p a r a l l e l. S i n c e space O space i s space t h e space m i d p o i n t comma space a n d space A C space a n d space O D space a r e space p a r a l l e l comma space b y space b a s i c space p r o p o r t i o n a l i t y space t h e o r e m comma space w e space h a v e comma fraction numerator A O over denominator O B end fraction equals fraction numerator C D over denominator D B end fraction rightwards double arrow C D equals D B space space left square bracket sin c e space O A equals O B right square bracket rightwards double arrow D space i s space t h e space m i d p o i n t space o f space t h e space s i d e space B C. W e space k n o w space t h a t space t h e space l i n e space j o i n i n g space t h e space m i d minus p o i n t s space o f space t w o space s i d e s space o f space a space t r i a n g l e space i s space p a r a l l e l space t o space t h e space t h i r d space s i d e space a n d space e q u a l space t o space h a l f space t h e space l e n g t h space o f space t h e space t h i r d space s i d e. T h u s comma space I n space triangle B A C comma space D space a n d space O space a r e space m i d p o i n t s space o f space t h e space s i d e s space A B space a n d space B C rightwards double arrow O D space i s space p a r a l l e l space t o space A C space a n d space O D equals 1 half cross times A C rightwards double arrow A C equals 2 cross times O D
Answered by Expert 22nd March 2015, 6:02 AM
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