A chord of length 16 cm is at a distance of 15 cm from the centre of the circle. The length of the chord of the same circle which is at a distance of 8 cm from the centre is

15cm

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16cm

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24cm

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30cm

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Solution

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

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