Question

Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

Answer 1

Given:

A circle with center O.

Diameter is AB

To Find:

Prove that the tangents drawn at the end of a diameter of a circle are parallel.

Solution:

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

Since PQ is the tangent at point A.

∴ OA ⊥ PQ

because tangents at any point of a circle is a perpendicular to the radius through the point of contact]

∠OAP = 90°  ..(i)

Similarly,

RS is a tangent on point B.

∴ OB ⊥ RS

because tangent at any point of a circle is perpendicular to the radius through the point of contact.

∠OBS = 90°  ..(ii)

Now, using (i) and (ii)

∠OAP = 90° and ∠OBS = 90°

This actively demonstrates that ∠OAP = ∠OBS

i.e. ∠BAP = ∠ABS

Foe angles PQ & RS,

and transversal AB

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

So, lines are parallel

∴ PQ ║ RS

Therefore, the tangents drawn at the end of a circle are parallel.