In the given figure, quadrilateral $A B C D$ is circumscribed and $A D ⊥ A B$ . If radius of incircle is $10 c m,$ then the value of $x$ is

20cm

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42cm

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19cm

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21cm

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Solution

The correct option is D 21cm
$G i v e n - O i s t h e c e n t r e o f a c i r c l e, i n s c r i b e d i n a q u a d r i l a t e r a l A B C D . T h e r a d i u s o f t h e c i r c l e i s 10 c m . ∠ B A D = 90^{o} . A B, B C, C D & A D t o u c h t h e i n s c r i b e d c i r c l e a t P, Q, R & S r e s p e c t i v e l y . C R = 27 c m a n d B C = 38 c m . T o f i n d o u t - A B = x = ? S o l u t i o n - W e j o i n O S & O P . T h e n O S & O P a r e r a d i i o f t h e i n s c r i b e d c i r c l e t h r o u g h t h e p o i n t s o f c o n t a c t o f t h e t a n g e n t s A S & A P r e s p e c t i v e l y . ∴ O S = O P = 10 c m . A g a i n ∠ O S A = 90^{o} = ∠ O P A s i n c e t h e r a d i u s t h r o u g h t h e p o i n t o f c o n t a c t o f a t a n g e n t t o a c i r c l e i s p e r p e n d i c u l a r t o t h e t a n g e n t . N o w i n O S A P, ∠ O S A = 90^{o} = ∠ O P A a n d O S = O P = 10 c m . ∴ O S A P i s a s q u a r e o f s i d e 10 c m . S o A P = O S = 10 c m . A g a i n C R = C Q = 27 c m a n d B Q = B P s i n c e t h e l e n g t h s o f t h e t a n g e n t s, f r o m a p o i n t t o a c i r c l e, a r e e q u a l . ∴ B Q = B C - C Q = (38 - 27) c m = 11 c m . A l s o B P = B Q = 11 c m . ∴ A B = x = A P + B P = (10 + 11) c m = 21 c m . A n s - O p t i o n D .$

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In the given figure, quadrilateral ABCD is circumscribed and AD⊥AB. If radius of incircle is 10cm, then the value of x is

In the given figure, quadrilateral $A B C D$ is circumscribed and $A D ⊥ A B$ . If radius of incircle is $10 c m,$ then the value of $x$ is