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Intersection between Tangent and Secant

In the figure...

Question

In the figure, $s e g A B$ and $s e g A D$ are the chords of the circle. $C$ is a point on the tangent of the circle at point $A$ . If $m (a r c A P B) = 80^{0}$ and $∠ B A D = 30^{0}$ , m $∠ B A C$ in degree is

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Solution

The correct option is C 40
$G i v e n - O i s t h e c e n t r e o f a c i r c l e w i t h a t a n g e n t A C a t A . A B & A D a r e t w o c h o r d s . m (a r c A P B) = 60^{o} a n d ∠ B A D = 30^{o} . T o f i n d o u t - ∠ B A C = ? . S o l u t i o n - W e j o i n A O, B O a n d B D . N o w m (a r c A P B) = 80^{o} m e a n s ∠ A O B = 80^{o} . W e k n o w t h a t t h e a n g l e, s u b t e n d e d b y a c h o r d o f a c i r c l e a t t h e c e n t r e, i s d o u b l e t h e a n g l e s u b t e n d e d b y t h e s a m e c h o r d a t t h e c i r c u m f e r e n c e o f t h e c i r c l e . ∴ ∠ A D B = \frac{1}{2} \times ∠ A O B = \frac{1}{2} \times 80^{o} = 40^{o} . A g a i n w e k n o w t h a t t h e a n g l e, b e t w e e n a t a n g e n t t o a c i r c l e a n d t h e c h o r d a t t h e p o i n t o f c o n t a c t, i s e q u a l t o t h e a n g l e s u b t e n d e d b y t h e s a m e c h o r d i n t h e a l t e r n a t e s e g m e n t o f t h e c i r c l e . ∴ ∠ B A C = ∠ A D B = 40^{o} . A n s - O p t i o n C .$

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In the figure, segAB and segAD are the chords of the circle. C is a point on the tangent of the circle at point A. If m(arcAPB)=800 and ∠BAD=300, m∠BAC in degree is

In the figure, $s e g A B$ and $s e g A D$ are the chords of the circle. $C$ is a point on the tangent of the circle at point $A$ . If $m (a r c A P B) = 80^{0}$ and $∠ B A D = 30^{0}$ , m $∠ B A C$ in degree is