# In Fig., tangents

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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