In a circle of radius 17cm two parallel chords of the length 30cm and 16cm respectively are drawn on the opposite sides of the centre. Then the distance between them is

7cm

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12cm

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23cm

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32cm

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Solution

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

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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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