If a line intersects, two concentric circles (circles with the same centre) with centre O at A, B, C and D, Then the relation between AB & CD is

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AB=CD

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AB>CD

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NO RELATION

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Solution

The correct option is B AB=CD
$G i v e n - O i s t h e c e n t r e o f t w o c o n c e n t r i c c i r c l e s . T h e l i n e ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ A B C D i n t e r s e c t s t h e o u t e r c i r c l e a t A & D a n d t h e i n n e r c i r c l e a t B & C . T o f i n d o u t t h e v a l i d i t y o f t h e a s s e r t i o n A B = C D S o l u t i o n - W e j o i n O A, O D, O B, O C . A l s o w e d r o p a p e r p e n d i c u l a r O N f r o m O t o ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ A B C D . O N m e e t s ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ A B C D a t N . N o w O N ⊥ A D a t N . ∴ N i s t h e m i d p o i n t o f A D s i n c e t h e p e r p e n d i c u l a r, d r o p p e d f r o m t h e c e n t e r o f a c i r c l e t o i t s a n y c h o r d b i s e c t s t h e l a t t e r . ∴ A N = N D . . . . (i) S i m i l a r l y O N ⊥ B C a t N . ∴ N i s t h e m i d p o i n t o f B C s i n c e t h e p e r p e n d i c u l a r, d r o p p e d f r o m t h e c e n t e r o f a c i r c l e t o i t s a n y c h o r d b i s e c t s t h e l a t t e r . ∴ B N = N C . . . . (i i) . S u b t r a c t i n g (i i) f r o m (i) w e g e t A N - B N = N D - N C ⟹ A B = C D . A n s O p t i o n B .$

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O

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In a figure, If a line intersects two concentric circles (circles with the same centre) with centre O at A, B, C and D. Prove that AB = CD

If a line intersects two concentric circles (circles with the same centre) with centre O at A, B, C and D, prove that AB = CD (see figure 10.25).

Question 4
If a line intersects two concentric circles (circles with the same centre) with centre O at A, B, C and D, prove that AB = CD.

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