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AB is the diameter of circle with center O.

Chord CD is equal to radius OC. AC and BOD produced intersect at P .

Find:

OCD

APB

Asked by mythili Suresh | 12th Sep, 2014, 07:43: PM

Expert Answer:

$C o n s i d e r space t h e space f o l l o w i n g space f i g u r e.$

$open parentheses 1 close parentheses A B space i s space t h e space d i a m t e r space o f space t h e space c i r c l e. G i v e n space t h a t space C D equals r a d i u s space o f space t h e space c i r c l e equals O A equals O C equals O D equals C D 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 O C D S i n c e space O C equals O D equals C D comma space triangle O C D 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 A n d space h e n c e space angle O C D equals angle D O C equals angle C D O equals 60 degree open parentheses 2 close parentheses 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 s space triangle C A O space a n d space triangle B D O S i n c e space O A equals O C equals O D equals O B space a n d space triangle O C D 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 comma space w e space h a v e triangle C A O space a n d space triangle B D O space a r e space a l s o space e q u i l a t e r a l space t r i a n g l e s space a n d space h e n c e comma space A C equals B D equals C D W E space h a v e comma space A C equals O D space a n d space O A equals C D comma space t h e r e f o r e comma space O A C D space i s space a space p a r a l l e l o l g r a m T h e space d i a g o n a l space o f space t h e space p a r a l l e log r a m space b i s e c t s space t h e space a n g l e space O A C space a n d space h e n c e angle C A D equals 1 half angle C O D equals 1 half cross times 60 degree equals 30 degree A n g l e space i n space a space s e m i c i r c l e space i s space 90 degree T h u s comma space angle A D B equals 90 degree rightwards double arrow angle A D P equals 90 degree C o n s i d e r space t h e space t r i a n g l e space triangle A D P 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 angle A D P plus angle D P A plus angle D A P equals 180 degree rightwards double arrow 90 degree plus angle D P A plus 30 degree equals 180 degree rightwards double arrow angle D P A equals 180 degree minus 120 degree equals 60 degree$

Answered by Vimala Ramamurthy | 14th Sep, 2014, 12:10: PM

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