$O$ is the centre of the circle with radius $5 c m$ . Chords $A B$ and $C D$ are parallel. $A B = 6 c m$ and $C D = 8 c m$ . If $P Q$ is distance between $A B$ and $C D$ , then $P Q$ is equal to

10 c m

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8 c m

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7 c m

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7 \sqrt{2} c m

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Solution

The correct option is A $7 c m$
$G i v e n - O i s t h e c e n t r e o f a c i r c l e o f r a d i u s 5 c m . A B = 6 c m & C D = 8 c m a r e p a r a l l e l c h o r d s w h o a r e a t p e r p e n d i c u l a r d i s t a n c e o f P Q . T o f i n d o u t - P Q = ? S o l u t i o n - W e j o i n O C a n d d r o p p e r p e n d i c u l a r s O M & O N f r o m O t o A B & C D . O M & O N m e e t A B & C D a t M & N r e s p e c t i v e l y . ∴ M & N a r e m i d p o i n t s o f A B & C D r e s p e c t i v e l y s i n c e t h e p e r p e n d i c u l a r, d r o p p e d f r o m t h e c e n t e r o f a c i r c l e t o a n y o f i t s c h o r d b i s e c t s t h e l a t t e r . S o A M = \frac{1}{2} \times A B = \frac{1}{2} \times 6 c m = 3 c m a n d C N = \frac{1}{2} \times C D = \frac{1}{2} \times 8 c m = 4 c m . N o w O M ⊥ A B & O N ⊥ C D . ∴ ∠ O M A = 90^{o} = ∠ O N C . i . e Δ O M A & Δ O N C a r e r i g h t t r i a n g l e s w i t h O A & O C a s h y p o t e n u s e s r e s p e c t i v e l y . S o, b y P y t h a g o r a s t h e o r e m, w e h a v e O M = \sqrt{{O A}^{2} - {A M}^{2}} = \sqrt{5^{2} - 3^{2}} c m = 4 c m a n d O N = \sqrt{{O C}^{2} - {C N}^{2}} = \sqrt{5^{2} - 4^{2}} c m = 3 c m . ∴ M N = O M + O N = (4 + 3) c m = 7 c m . N o w A B ∥ C D a n d P Q i s t h e p e r p e n d i c u l a r d i s t a n c e b e t w e e n t h e m . A l s o O M & O N a r e o n t h e s a m e s t r a i g h t l i n e a s b o t h o f t h e m a r e p e r p e d i c u l a r t o a p a i r o f s t r a i g h t l i n e s A B & C D a n d t h e y p a s s t h r o u g h t h e s a m e p o i n t O . S o M N i s t h e p e r p e n d i c u l a r d i s t a n c e b e t w e e n A B & C D . ∴ P Q = M N = 7 c m . A n s - O p t i o n C .$

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O is the centre of the circle with radius 5 cm . Chords AB and CD are parallel. AB=6 cm and CD=8 cm. If PQ is distance between AB and CD, then PQ is equal to

$O$ is the centre of the circle with radius $5 c m$ . Chords $A B$ and $C D$ are parallel. $A B = 6 c m$ and $C D = 8 c m$ . If $P Q$ is distance between $A B$ and $C D$ , then $P Q$ is equal to