A diameter of a circle bisects each of the two chords of the circle. Then the chords are (write the relative orientation)

at any angle .

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perpendicular to each other.

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parallel to each other.

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at an obtuse angle.

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Solution

The correct option is B parallel to each other.
$G i v e n - A B & C D a r e t w o c h o r d s o f a c i r c l e w i t h c e n t r e O . T h e d i a m e t e r P Q o f t h e g i v e n c i r c l e b i s e c t s A B a t M a n d C D a t N . T o f i n d o u t - T h e r e l a t i v e o r i e n t a t i o n o f A B & C D = ? S o l u t i o n - P Q i s a d i a m e t e r o f t h e g i v e n c i r c l e . S o P Q w i l l p a s s t h r o u g h O . ∴ A s p e r t h e g i v e n c o n d i t i o n, O M b i s e c t s A B a t M . i . e O M ⊥ A B . ∴ ∠ O M A = 90^{o} = ∠ O M B . A l s o O N b i s e c t s C D a t N . i . e O N ⊥ C D . S i n c e i f a l i n e t h r o u g h t h e c e n t r e o f a c i r c l e b i s e c t s a c h o r d o f t h e s a m e c i r c l e, t h e n t h e l i n e w i l l b e p e r p e n d i c u l a r t o t h e s a m e c h o r d . ∠ O N C = 90^{o} = ∠ O N D . ∴ ∠ O M A = ∠ O N D . B u t t h e y a r e a l t e r n a t e a n g l e s . ∴ A B ∥ C D . S o t h e c h o r d a r e p a r a l l e l t o e a c h o t h e r . A n s - O p t i o n C .$

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If a diameter of a circle bisects each of the two chords of a circle then prove that the chords are parallel.

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