Two parallel chords of lengths 30 cm and 16 cm are drawn on the opposite sides of the centre of a circle of a radius 17 cm. The distance between the chords is

32cm

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23cm

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46cm

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16cm

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Solution

The correct option is B 23cm
$G i v e n - O i s t h e c e n t r e o f a c i r c l e o f r a d i u s 17 c m . A B = 16 c m & C D = 30 c m a r e p a r a l l e l c h o r d s w h o a r e a t p e r p e n d i c u l a r d i s t a n c e o f P Q . T o f i n d o u t - P Q = ? S o l u t i o n - W e j o i n O C a n d d r o p p e r p e n d i c u l a r s O M & O N f r o m O t o A B & C D . O M & O N m e e t A B & C D a t M & N r e s p e c t i v e l y . ∴ M & N a r e m i d p o i n t s o f A B & C D r e s p e c t i v e l y s i n c e t h e p e r p e n d i c u l a r, d r o p p e d f r o m t h e c e n t e r o f a c i r c l e t o a n y o f i t s c h o r d b i s e c t s t h e l a t t e r . S o A M = \frac{1}{2} \times A B = \frac{1}{2} \times 16 c m = 8 c m a n d C N = \frac{1}{2} \times C D = \frac{1}{2} \times 30 c m = 15 c m . N o w O M ⊥ A B & O N ⊥ C D . ∴ ∠ O M A = 90^{o} = ∠ O N C . i . e Δ O M A & Δ O N C a r e r i g h t t r i a n g l e s w i t h O A & O C a s h y p o t e n u s e s r e s p e c t i v e l y . S o, b y P y t h a g o r a s t h e o r e m, w e h a v e O M = \sqrt{{O A}^{2} - {A M}^{2}} = \sqrt{17^{2} - 8^{2}} c m = 15 c m a n d O N = \sqrt{{O C}^{2} - {C N}^{2}} = \sqrt{17^{2} - 15^{2}} c m = 8 c m . ∴ M N = O M + O N = (15 + 8) c m = 23 c m . N o w A B ∥ C D a n d P Q i s t h e p e r p e n d i c u l a r d i s t a n c e b e t w e e n t h e m . A l s o O M & O N a r e o n t h e s a m e s t r a i g h t l i n e a s b o t h o f t h e m a r e p e r p e d i c u l a r t o a p a i r o f s t r a i g h t l i n e s A B & C D a n d t h e y p a s s t h r o u g h t h e s a m e p o i n t O . S o M N i s t h e p e r p e n d i c u l a r d i s t a n c e b e t w e e n A B & C D . ∴ P Q = M N = 23 c m . A n s - O p t i o n B .$

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In a circle of radius 17 cm, two parallel chords of lengths 30 cm and 16 cm are drawn.

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