Q. 74.4( 20 Votes )
In figure, tangents PQ and PR are drawn to a circle such that ∠RPQ = 30°. A chord RS is drawn parallel to the tangent PQ. Find the ∠RQS.
Answer :
Given : Tangents PQ and PR are drawn to a circle such that ∠RPQ = 30°. A chord RS is drawn parallel to the tangent PQ.
To Find : ∠RQS
PQ = PR [Tangents drawn from an external point to a circle are equal]
∠PRQ = ∠PQR [Angles opposite to equal sides are equal] [1]
In △PQR
∠PRQ + ∠PQR + ∠QPR = 180°
∠PQR + ∠PQR + ∠QPR = 180° [Using 1]
2∠PQR + ∠RPQ = 180°
2∠PQR + 30 = 180
2∠PQR = 150
∠PQR = 75°
∠QRS = ∠PQR = 75° [Alternate interior angles]
∠QSR = ∠PQR = 75° [angle between tangent and the chord equals angle made by the chord in alternate segment]
Now In △RQS
∠RQS + ∠QRS + ∠QSR = 180
∠RQS + 75 + 75 = 180
∠RQS = 30°
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PREVIOUSIn a right angle ΔABC is which ∠B = 90°, a circle is drawn with AB as diameter intersecting the hypotenuse AC at P. Prove that the tangent to the circle at PQ bisects BC.NEXTAB is a diameter and AC is a chord of a circle with center O such that ∠BAC = 30°. The tangent at C intersects extendedAB at a point D. Prove that BC = BD.
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