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Q.

The angle between the two tangents drawn from an external point to a circle is ____ to the angle subtended by the line segments joining the points of contact at the centre.

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Detailed Solution

The angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segments joining the points of contact at the centre.
We know that sum of all the angles of a quadrilateral is 360° and the angles around a Point Perpendicular bisector
Given that

PA

and

PB

are the tangents drawn from a point

P

to a circle with center

O

and also the line segments

OA

and

OB

that are drawn.
Question Image

Question Image

Using the perpendicular bisector theorem, the perpendicular bisector of a chord passes through the center of the circle.

PA ⊥ OA ∠ OAP = 90 ° PB ⊥ OB ∠ OBP = 90 ° ∠ OAP + ∠ OBP = 90 ° + 90 ° ∠ OAP + ∠ OBP = 180 ° . … … i

Using the angles of a quadrilateral property that states, the sum of all the angles of a quadrilateral is 360°.

∠ OAP + ∠ OBP + ∠ APB + ∠ AOB = 360 ° … … (ii)

[From (i) and (ii)]

∠ APB + ∠ AOB = 180 °

Thus, the angles are supplementary.

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The angle between the two tangents drawn from an external point to a circle is ____ to the angle subtended by the line segments joining the points of contact at the centre.