Q. 174.4( 7 Votes )

In Fig., tangents

Answer :

Given: RPQ=30°


tangents PQ and PR are drawn from an external point P.


To find: RSQ


Theorem Used:


1.) The length of two tangents drawn from an external point are equal.


2.) Tangent to a circle at a point is perpendicular to the radius through the point of contact.


Explanation:



As P is external point and PR and PQ are tangents,


By theorem (1) stated above,


PQ = PR


As angles opposite to equal sides are equal.


RQP = QRP


So PQR is an isosceles triangle.


By angle sum property,



RQP+QRP+RPQ=180°



2RQP+30° =180°


2RQP=150°


⇒∠RQP=QRP=75°


From the theorem (2) stated above,


OQP = ORP = 90°


We know sum of angles of a quadrilateral is equal to 360°.


OQP + ORP + RPQ + QOR = 360°


90° + 90° + 30° + QOR = 360°


210° + QOR = 360°


QOR = 150°


As we know angle subtended by an arc at any point on the circle is half the angle subtended at the centre by the same arc,



Also QSR = SQU (alternate angles)


SQU = 75°


Angle on a straight line is 180°,


So,


SQU + RQS + PQR = 180°


75° + RQS + 75° = 180°


150° + RQS = 180°



RQS = 30°




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