Question

show that the tangents drawn at the ends of a diameter of a circle are parallei.

Answer 1

you know that tangents to a circle subtend a right angle on the radius of the circle at the point of intersection with the circumference. here, we are looking at the tangents that are diametrically opposite. they both are perpendicular to the diameter which is a straight line. hence they are both perpendicular to the same straight line which directly implies that they are parallel.

Answer 2

Let AB be a diameter of the circle. Two tangents PQ and RS are drawn at points A and B respectively.

Radius drawn to these tangents will be perpendicular to the tangents.

Thus, OA ⊥ RS and OB ⊥ PQ

∠OAR = 90º

∠OAS = 90º

∠OBP = 90º

∠OBQ = 90º

It can be observed that

∠OAR = ∠OBQ (Alternate interior angles)

∠OAS = ∠OBP (Alternate interior angles)

Since alternate interior angles are equal, lines PQ and RS will be parallel