Question
Mon March 09, 2015 By: Sujay Sarkar

If two intersecting chors of a circle make equal angles with the diameter passing through their point of intersection, prove that the chords are equal.

Expert Reply
Vimala Ramamurthy
Wed March 11, 2015
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.
C o n s i d e r space a space c i r c l e space w i t h space c e n t r e space a t space O. L e t space A B space a n d space C D space b e space t h e space i n t e r s e c t i n g space c h o r d s space w h i c h space m a k e space e q u a l space a n g l e s space w i t h space t h e space d i a m e t e r space P R space p a s sin g t h r o u g h space Q. W e space n e e d space t o space p r o v e space t h a t space A B equals C D L e t space M space a n d space N space a r e space t h e space f o o t space o f space p e r p e n d i c u l a r s space f r o m space t h e space c e n t r e space t o space t h e space c h o r d s. N o w space c o n s i d e r space triangle O Q M space a n d space triangle O Q N angle O M Q equals angle O N Q equals 90 degree angle O Q M equals angle O Q N space space space space space open square brackets G i v e n close square brackets O Q equals O Q space space space space space space space space space space space space space space space space space open square brackets c o m m o n close square brackets T h u s comma space b y space A n g l e minus A n g l 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 triangle O Q M approximately equal to triangle O Q N 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. T h u s comma space O M equals O N. W e space k n o w space t h a t space e q u i d i s tan t space c h o r d s space f r o m 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 r e space e q u a l space i n space l e n g t h T h u s comma space w e space h a v e comma A B equals C D
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