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Tangent Theorem

In the figure...

Question

In the figure, $P Q$ is a diameter of a circle with centre at $O$ and $O R ⊥ P Q$ , where $R$ is a point on the circle. If $S$ is another point on the circle such that $∠ R P S = 32^{\circ}$ , then $∠ Q R S$ is

13^{\circ}

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26^{\circ}

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45^{\circ}

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30^{\circ}

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Solution

The correct option is A $13^{\circ}$
$G i v e n - P Q i s t h e d o a m e t e r o f a c i r c l e w i t h c e n t r e O . R & S a r e t w o p o i n t s o n t h e c i r c u m f e r e n c e o f t h e g i v e n c i r c l e s u c h t h a t O R ⊥ P Q & ∠ R P S = 32^{o} . T o f i n d o u t - ∠ Q R S = ? S o l u t i o n - W e j o i n R Q & R S . I n Δ O P R O P = O R (r a d i i o f t h e s a m e c i r c l e) . S o Δ O P R i s a n i s o s c e l e s o n e w i t h P R a s b a s e a n d ∠ P O R = 90^{o} (s i n c e O R ⊥ P Q) . ∴ ∠ O P R = ∠ O R P ⟹ ∠ O P R + ∠ O R P = 2 ∠ O P R . S o 2 ∠ O P R = 180^{o} - ∠ P O R ⟹ 2 ∠ O P R = 180^{o} - 90^{o} = 90^{o} ⟹ ∠ O P R = 45^{o} . ∴ ∠ Q P S = ∠ O P R - ∠ P R S = 45^{o} - 32^{o} = 13^{o} . N o w t h e c h o r d Q S s u b t e n d s ∠ Q P S & ∠ Q R S t o t h e c i r c u m f e r e n c e o f t h e g i v e n c i r c l e a t P & R r e s p e c t i v e l y . ∴ ∠ Q P S = ∠ Q R S = 13^{o} (s i n c e t h e a n g l e s, s u b t e n d e d b y a c h o r d o f a c i r c l e t o i t s c i r c u m f e r e n c e, a r e e q u a l) . A n s - O p t i o n A .$

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Similar questions

Q. Suppose Q is a point on the circle With centre P and radius 1, as shown in the figure; R is a point outside the circle such that QR = 1 and

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In figure, PQ is a tangent from an external point P to a circle with centre O and OP cuts the circle at T and QOR is a diameter. If $∠ P O R = 130^{\circ}$ and S is a point on the circle, find $∠ 1 + ∠ 2$

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In the figure, PQ is a diameter of a circle with centre at O and OR⊥PQ, where R is a point on the circle. If S is another point on the circle such that ∠RPS=32∘, then ∠QRS is

In the figure, $P Q$ is a diameter of a circle with centre at $O$ and $O R ⊥ P Q$ , where $R$ is a point on the circle. If $S$ is another point on the circle such that $∠ R P S = 32^{\circ}$ , then $∠ Q R S$ is