These two circles APC and BPC are equal. They intersect at points P and C, a st. line passes through A,P and B. Now, join AC and BC. Prove that AC = BC.

Asked by Apoorv | 7th Nov, 2014, 10:45: PM

Expert Answer:

S i n c e space t h e space c i r c l e s space a r e space e q u a l comma space t h e space r a d i i space a r e space a l s o space e q u a l. I f space A C space i s space t h e space d i a m e t e r space o f space t h e space c i r c l e comma space t h e n space t h e space s o l u t i o n space i s space v e r y space t r i v i a l comma space t h a t space i s space A C equals B C. L e t space O subscript 1 space a n d space O subscript 2 space b e space t h e space c e n t r e s space o f space t h e space c i r c l e s space A P C space a n d space B P C space r e s p e c t i v e l y.
L e t space angle A P C equals x space a n d space angle B P C equals y T h e space a n g l e space f o r m e d space a t 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 i s space d o u b l e space t h e space a n g l e space f o r m e d space o n space t h e space c i r c u m f e r e n c e space o f space t h e space c i r c l e. T h e r e f o r e comma space angle A O subscript 1 C equals 2 x space a n d space angle B O subscript 2 C equals 2 y A l s o space P C space i s space perpendicular O subscript 1 O subscript 2 space a n d space O subscript 1 O subscript 2 parallel to A B. T h u s comma space w e space h a v e comma space x equals 90 degree space a n d space y equals 90 degree T h e r e f o r e comma space a n g l e s comma space space angle A O subscript 1 C space a n d space angle B O subscript 2 C space a r e space s t r a i g h t space a n g l e s space a n d space h e n c e space t h e y space c o i n c i d e space w i t h space t h e space l i n e s space A C space a n d space B C. S i n c e space t h e y space a r e space o f f space e q u a l space c i r c l e s comma space d i a m e t e r s space a r e space a l s o space e q u a l space a n d space h e n c e comma space A C equals B C.

Answered by Vimala Ramamurthy | 10th Nov, 2014, 10:31: AM

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