if 2 equal chords of a circle intersect within the circle intersect within the circle. Prove that the line joining the points of intersection to the centre makes equal angles with the chords.

Asked by shub3893 | 3rd Dec, 2015, 08:47: 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.
A B space a n d space C D space a r e space e q u a l space c h o r d s space a n d space t h e y space i n t e r s e c t space a t space p o i n t space P. L i n e space j o i n i n g space t h e space c e n t r e space a n d space t h e space c h o r d space i s space a space p e r p e n d i c u l a r space b i s e c t o r space o f space t h e space c h o r d. T h u s comma space O R space a n d space O S space a r e space p e r p e n d i c u l a r space b i s e c t o r s space o f space t h e space c h o r d s comma space A B space a n d space C D space r e s p e c t i v e l y.  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 s space O P R space a n d space O P S colon angle O R P equals angle O S P equals 90 degree O P equals O P space space space space open square brackets c o m m o n space h y p o t e n u s e close square brackets O R equals O S space space space space open square brackets because e q u a l space c h o r d s space a r e space e q u i d i s tan t space f r o m space t h e space c e n t r e close square brackets T h u s comma space b y space R i g h t space a n g l e minus H y p o t e n u s e minus S i d e space c r i t e r i o n space o f space c o n g r u e n c e comma space w e space h a v e comma increment O P R space approximately equal to increment space O P S T h e space c o r r e s p o n d i n g space p a r t s space o f space t h e space c o n g r u e n t space t r i a n g l e s space a r e space c o n g r u e n t. rightwards double arrow angle O P R equals angle O P S rightwards double arrow T h e space l i n e space j o i n i n g space t h e space p o i n t s space o f space i n t e r s e c t i o n space t o space t h e space c e n t r e space m a k e s space e q u a l space a n g l e s space w i t h space t h e space c h o r d s.

Answered by Vimala Ramamurthy | 4th Dec, 2015, 11:50: AM