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If two chords AB and CD ofa a circle AYDZBWCX at right angles, prove that arc CXA + arc DZB = arc AYD + arc BWC = semicircle. 

Asked by ashokkushwaha12 27th February 2015, 1:26 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.
O space i s space a n y space p o i n t space i n s i d e space t h e space c i r c l e. N o w space c o n s i d e r space t h e space c h o r d space C A. T h e space a n g l e space s u b t e n d e d space b y space t h e space c h o r d space A C space a t space t h e space c i r c u m f e r e n c e space i s space angle C B A S i m i l a r l y space t h e space a n g l e space s u b t e n d e d space b y space t h e space c h o r d space B D space a t space t h e space c i r c u m f e r e n c e space i s space angle B C D N o w space 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 comma space B O C. T h u s comma space 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 C O B plus angle C B A plus angle B C D equals 180 degree rightwards double arrow 90 degree plus angle C B A plus angle B C D equals 180 degree rightwards double arrow angle C B A plus angle B C D equals 180 degree minus 90 degree rightwards double arrow angle C B A plus angle B C D equals 90 degree T h a t space i s space t h e space s u m space o f space a n g l e space s u b t e n d e d space b y space t h e space a r c space C X A space a n d space t h e a n g l e space s u b t e n d e d space b y space t h e space a r c space B Z D equals 90 degree a r c stack C X A with overparenthesis on top plus a r c stack B Z D with overparenthesis on top equals 90 degree.... left parenthesis 1 right parenthesis  N o w space c o n s i d e r space t h e space c h o r d space B C. T h e space a n g l e space s u b t e n d e d space b y space t h e space c h o r d space B C space a t space t h e space c i r c u m f e r e n c e space i s space angle B A C S i m i l a r l y space t h e space a n g l e space s u b t e n d e d space b y space t h e space c h o r d space A D space a t space t h e space c i r c u m f e r e n c e space i s space angle A C D N o w space 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 comma space A O C. T h u s comma space 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 C O A plus angle B A C plus angle A C D equals 180 degree rightwards double arrow 90 degree plus angle B A C plus angle A C D equals 180 degree rightwards double arrow angle B A C plus angle A C D equals 180 degree minus 90 degree rightwards double arrow angle B A C plus angle A C D equals 90 degree T h a t space i s space t h e space s u m space o f space a n g l e space s u b t e n d e d space b y space t h e space a r c space C W B space a n d space t h e a n g l e space s u b t e n d e d space b y space t h e space a r c space A Y D equals 90 degree a r c stack C W B with overparenthesis on top plus a r c stack A Y D with overparenthesis on top equals 90 degree.... left parenthesis 2 right parenthesis F r o m 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 a r c stack C X A with overparenthesis on top plus a r c stack B Z D with overparenthesis on top equals a r c stack C W B with overparenthesis on top plus a r c stack A Y D with overparenthesis on top equals 90 degree W e space k n o w space t h a t space t h e space a r c space o f space a space c i r c l e space s u b t e n d i n g space a space r i g h t space a n g l e space a t space a n y space p o i n t space o f space t h e c i r c l e space i n space i t s space a l t e r n a t e space s e g m e n t space i s space a space s e m i space c i r c l e. T h u s comma space w e space h a b e comma a r c stack C X A with overparenthesis on top plus a r c stack B Z D with overparenthesis on top equals a r c stack C W B with overparenthesis on top plus a r c stack A Y D with overparenthesis on top equals s e m i space c i r c l e
Answered by Expert 28th February 2015, 9:31 AM
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