Construct a tangent to a circle of radius $3 c m$ from a point on the concentric circle of radius $5 c m$ and measure its length. Also verify the measurement by actual calculation.

5 c m, v e r i f i e d

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4 c m, v e r i f i e d

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5 c m, c a n^{'} t b e v e r i f i e d

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3 c m, v e r i f i e d

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Solution

The correct option is B $4 c m, v e r i f i e d$
$T o c o n s t r u c t - t w o c o n c e n t r i c c i r c l e s w i t h r a d i i 3 c m & 5 c m a n d t o d r a w a t a n g e n t f r o m a p o i n t A o n t h e c i r c u m f e r e n c e o f t h e o u t e r c i r l e t o t h e i n n e r c i r c l e . A l s o t o m e a s u r e t h e l e n g t h o f t h i s t a n g e n t a n d j u s t i f y t h e r e s u l t b y c a l c u l a t i o n . C o n s t r u c t i o n - (I) T a k i n g O a s c e n t r e a n d r a d i i 3 c m & 5 c m, t w o c o n c e n t r i c c i r c l e s a r e d r a w n . (I I) W e t a k e a n y p o i n t A o n t h e c i r c u m f e r e n c e o f t h e o u t e r c i r c l e a n d j o i n A O . (I I I) T a k i n g A O a s d i a m e t e r, a s e m i c i r c l e A P B i s d r a w n o n A O . T h e s e m i c i r c l e A P B i n t e r s e c t s t h e i n n e r c i r c l e a t P . (I V) A P i s j o i n e d . (V) A P i s m e a s u r e d . A P = 4 c m . J u s t i f i c a t i o n - O P i s a r a d i u s o f t h e g i v e n i n n e r c i r c l e a n d i t i s a n a n g l e i n t h e s e m i c i r c l e A P B . ∴ ∠ O A P = 90^{o} ⟹ O P ⊥ A P . N o w w e k n o w t h a t h e r a d i u s, t h r o u g h t h e p o i n t o f c o n t a c t o f a t a n g e n t t o a c i r c l e, i s p e r p e n d i c u l a r t o t h e t a n g e n t . ∴ A P i s a t a n g e n t t o t h e g i v e n i n n e r c i r c l e . ∴ A P i s t h e r e q u i r e d t a n g e n t . A g a i n Δ O A P i s a r i g h t o n e w i t h O A a s h y p o t e n u s e s i n c e ∠ O A P = 90^{o} . S o, b y P y t h a g o r a s t h e o r e m, A P = \sqrt{{O A}^{2} - {O P}^{2}} = \sqrt{5^{2} - 3^{2}} c m = 4 c m . S o, t h e l e n g t h o f t h e t a n g e n t = A P = 4 c m b y a c t u a l m e a s u r e m e n t a s w e l l a s b y l o g i c a l a r g u m e n t s . A n s - O p t i o n B .$

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