Q) Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.

9th Class Maths – NCERT Important Questions

Ans:

Step 1:

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

Here, PT is the tangent to a circle with center O.

We have drawn QT as Ʇ PT and considered that it is not passing thru center O but thru point Q.

We need to prove that perpendicular to tangent PT passes thru center O.

Step 2:
∵ QT is Ʇ PT

∴  ∠ QTP = 90 ⁰

But, we know that radius is Ʇ to tangent.

Here OT is radius and PT is tangent

∴  ∠ OTP = 90 ⁰

Step 3:

By comparing above relation, we see that

∠ QTP = ∠ OTP = 90 ⁰ 

Since this can only be possible if Point Q lies on point O, and QT lies on OT.

∴ Line QT, if perpendicular to tangent PT, will passes thru center O.

Therefore, the point of intersection of the circles lies on the third side…… Hence Proved !

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