O is the centre of the circle of the circle. BC is a diameter of the circle. $O D ⊥ A B$ (chord). If OD= 4 cm, BD = 5 cm, then CD=

13 cm

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\sqrt{71} c m

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\sqrt{89} c m

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None of these

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Solution

The correct option is D None of these
$G i v e n - B C i s a d i a m e t e r o f t h e c i r c l e w i t h c e n t r e O . A B i s a n o t h e r c h o r d o f t h e s a m e c i r c l e . O D ⊥ A B . O D = 4 c m & B D = 6 c m . C D h a s b e e n j o i n e d . T o f i n d o u t - C D = ? W e j o i n A C . O D ⊥ A B ⟹ ∠ B D O = 90^{o} . i . e Δ B D O i s a r i g h t o n e w i t h O B a s h y p o t e n u s e . S o, b y P y t h a g o r a s t h e o r e m, w e g e t O B = \sqrt{{B D}^{2} + {O D}^{2}} = \sqrt{6^{2} + 4^{2}} c m = \sqrt{52} c m . N o w O B = O C = c m (r a d i i o f t e s a m e c i r c l e) . ∴ B C = O B + O C = (\sqrt{52} + \sqrt{52}) c m = 2 \sqrt{52} c m . A l s o D i s t h e m i d p o i n t o f A B s i n c e O D ⊥ A B a n d w e k n o w t h a t t h e p e r p e n d i c u l a r, f r o m t h e c e n t r e o f a c i r c l e t o a n y o f i t s c h o r d s, b i s e c t s t h e l a t t e r . ∴ A B = 2 B D = 2 \times 6 c m = 12 c m . A g a i n ∠ B A C = 90^{o} s i n c e i t i s a n a n g l e i n a s e m i c i r c l e = 90^{o} . ∴ Δ A B C i s a r i g h t o n e w i t h B C a s a s h y p o t e n u s e . S o, b y P y t h a g o r a s t h e o r e m, w e g e t A C = \sqrt{B C^{2} - A B^{2}} = \sqrt{{(\sqrt{52})}^{2} - 12^{2}} c m = 8 c m . N o w w e c n s i d e r Δ A D C . H e r e ∠ C A D = 90^{o} (a n g l e i n a s e m i c i r c l e = 90^{o}) . ∴ Δ A D C i s a r i g h t o n e w i t h C D a s a s h y p o t e n u s e . S o, b y P y t h a g o r a s t h e o r e m, w e g e t C D = \sqrt{A C^{2} + A D^{2}} = \sqrt{8^{2} + 6^{2}} c m = 10 c m . N o n e o f t h e o p t i o n s c o m p l y w i t h t h i s r e s u l t . A n s - O p t i o n D .$

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Similar questions

O

B C

O D ⊥ A B

O D = 4

B D = 5

C D

If BC is a diameter of a circle with centre O and OD is perpendicular to the chord AB of a circle, then CA = ____ OD.

In the given figure, AB is chord of a circle with centre O and BOC is a diameter. If OD $⊥$ AB such that OD = 6 cm then AC = ?

(a) 9 cm

(b) 12 cm

(d) 7.5 cm

O D ⊥ A B

O D = 6 cm

⊥

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