Question

18.
Prove that the tangents drawn at the end points of a diameter are parallel. ​

Answer 1

Step-by-step explanation:

Given: A circle with center O and diameter AB.

Let PQ be the tangent at point A & RS be the tangent at point B.

To Prove: PQ∥RS

Proof:

Since PQ is tangent at point A

OA⟂PQ (Tangent at any point of a circle is perpendicular to the radius through point of contact)

∠OAP = 90° ... (i)

Similarly

RS is a tangent at point B

OB⟂RS (Tangent at any point of a circle is perpendicular to the radius through point of contact)

∠OBS = 90° ... (ii)

From (i) and (ii)

∠OAP=90° and ∠OBS = 90°

Therefore,

∠OAP=∠OBS

i.e, ∠BAP=∠ABS

For lines PQ and RS and transversal AB

∠BAP=∠ABS i.e., both alternate angles are equal

So, lines are parallel

∴ PQ∥RS