Two chords  AB and CD intersect at P  inside the circle.Prove that the sum of the all angles subtended by the arcs AC and BD at the centre O is equal to twice the angle APC.

Asked by prasadkalya2 | 18th Nov, 2014, 10:01: 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.
T h e space a n g l e space s u b t e n d e d space b y space a n space a r c space o f space a space c i r c l e space a t space t h e space c e n t r e space i s space t w i c e space t h e space a n g l e space s u b t e n d e d space b y space i t space a t space a n y space p o i n t space o n space t h e space r e m a i n i n g space p a s t space o f space t h e space c i r c l e. S i n c e space a r c s space A C space a n d space B D space s u b t e n d s space r e s p e c t i v e l y comma space angle A O D space a n d space angle B O C space a t space t h e space c e n t r e comma space w e space h a v e comma angle A O D equals 2 angle A B D.... left parenthesis 1 right parenthesis a n d space angle B O C equals 2 angle B D C.... left parenthesis 2 right parenthesis A d d i n g space e q u a t i o n s space left parenthesis 1 right parenthesis space a n d space left parenthesis 2 right parenthesis comma space w e space h a v e comma angle A O D plus angle B O C equals 2 angle A B D plus 2 angle B D C rightwards double arrow angle A O D plus angle B O C equals 2 open parentheses angle A B D plus angle B D C close parentheses... left parenthesis 3 right parenthesis W e space k n o w space t h a t space e x t e r i o r space a n g l e space o f space a space t r i a n g l e space i s space e q u a l space t o space t h e space s u m space o f space i t s space o p p o s i t e space i n t e r i o r space a n g l e s. N o w space c o n s i d e r space t r i a n g l e space B C P : rightwards double arrow angle A P D equals angle B D P plus angle D B P... left parenthesis 4 right parenthesis F r o m space t h e space f i g u r e angle P B D equals angle A B D space a n d space angle P D B equals angle C D B T h e r e f o r e comma space f r o m space e q u a t i o n s space left parenthesis 3 right parenthesis space a n d space left parenthesis 4 right parenthesis comma space w e space h a v e comma angle A O D plus angle B O C equals 2 angle A P D

Answered by Vimala Ramamurthy | 20th Nov, 2014, 10:33: AM