Q) Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the centre. 

9th Class Maths – NCERT Important Questions

Ans:

Step 1:

Let’s make a diagram for better understanding of the question:

Here, PA and PB are the tangents drawn from Point P and ∠ APB is the angle between these 2 tangents.

Line Segment AB subtends angle ∠ AOB at center of the circle.

We need to prove that ∠ APB + ∠ AOB = 180 ⁰

Step 2:

∵ We know that the tangent is perpendicular to radius,

∴ ∠ OAP = 90 ⁰

and ∠ OBP = 90 ⁰

Step 3: We know that in a cyclic Quadrilateral, sum of all Angles is 360 ⁰

∵ Here, OAPB is a cyclic quadrilateral

∴ ∠ APB + ∠ OAP  +  ∠ AOB  +  ∠ OBP = 360 ⁰

∴ ∠ APB + 90 ⁰  +  ∠ AOB  +  90 ⁰ = 360 ⁰

∴  ∠ APB + ∠ AOB  + 180 ⁰  = 360 ⁰

∴  ∠ APB + ∠ AOB = 360 ⁰ – 180 ⁰

∴  ∠ APB + ∠ AOB = 180 ⁰

Therefore, the angle between the two tangents drawn from an external point to a circle is SUPPLEMENTARY to the angle subtended by the line segment joining the points of contact at the centre.

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