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

Question

Homi · Accepted Answer

Understand the Question

A tangent to a circle is perpendicular to the radius at the point of contact.
When two tangents are drawn at the opposite endpoints of a diameter, the angles made with the diameter are both $90^\circ$ .
If a transversal intersects two lines such that alternate interior angles are equal (or interior angles on the same side sum to $180^\circ$ ), the two lines are parallel.

Step 1 · Apply Tangent-Radius Perpendicularity

Let $O$ be the centre of the circle and $AB$ be a diameter. Let line $RS$ be the tangent at point $A$ and line $PQ$ be the tangent at point $B$ . Diagram 1

Since the radius is perpendicular to the tangent at the point of contact: $OA \perp RS \implies \angle OAR = 90^\circ$ $OB \perp PQ \implies \angle OBQ = 90^\circ$

Step 2 · Prove Lines are Parallel

Consider lines $RS$ and $PQ$ intersected by transversal $AB$ : $\angle OAR = \angle OBQ = 90^\circ$

Since these are equal alternate interior angles, the lines are parallel: $RS \parallel PQ$

Answer

Hence proved, the tangents drawn at the ends of a diameter of a circle are parallel.

Common Mistakes

Assuming Any Chord Works: Tangents are parallel only when drawn at the endpoints of a diameter (passing through the center), not for general chords.
Angle Pairing Confusion: If using alternate interior angles, pair $\angle OAR$ with $\angle OBQ$ ; if using consecutive interior angles on the same side, show that $\angle OAR + \angle OBP = 90^\circ + 90^\circ = 180^\circ$ .

Question 4

More questions in Exercise 10.2