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In figure, tangents PQ and PR are drawn from an external point P to a circle with centre O, such that $∠ RPQ = 30 °$ . A chord RS is drawn parallel to the tangent PQ. Find $∠ RQS$ .

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30 °

40 °

60 °

50 °

answer is A.

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Detailed Solution

Given that tangents PQ and PR are drawn from an external point P to a circle with centre O, such that

∠ RPQ = 30 °

.
A chord RS is drawn parallel to the tangent PQ.
Question Image

As the tangents drawn from the external point to a circle are equal.
From the figure, PR and PQ are the tangents.
So, PR = PQ.
Also,

∠ P R Q = ∠ P Q R

, because angles opposite to equal sides are equal in a triangle.
In the

∆

PQR,
Sum of all the angles of a triangle is 180°.

∠ P R Q + ∠ P Q R + ∠ R P Q = 1800 ⇒ 2 ∠ P Q R + 300 = 1800 ⇒ ∠ P Q R = 15002 ⇒ ∠ P Q R = 750

So,

∠ P R Q = ∠ P Q R = 750

.
By alternate segment theorem,

∆

QRS is an isosceles triangle.
So,

∠ S R Q = ∠ R S Q = 750

By angle sum property in

∆

QRS,

\Rightarrow ∠ QRS + ∠ RSQ + ∠ RQS = 180 °

\Rightarrow 75 ° + 75 ° + ∠ RQS = 180 °

\Rightarrow ∠ RQS = 180 ° - 150 °

\Rightarrow ∠ RQS = 30 °

So, the value of

∠ RQS

30 °

.
Hence, the correct option is 1.

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