Question

prove that the tangents drawn at the end of a diameter of a circle are parallel ​

Answer 1

To prove: The tangents drawn at the ends of the diameter of a circle are parallel. (XY ║ PQ)

Proof:

(Diagram for reference attached below.)

We know that the radius of a circle is always perpendicular to the tangent.

Therefore,

OA ⊥ XY

⇒ ∠OAX = ∠OAY = 90°

Also,

OB ⊥ PQ

⇒ ∠OBP = ∠OBQ = 90°

A pair of alternate interior angles are equal.

∴ XY ║ PQ.

Answer 2

Heyya Buddy ^ - ^

Step-by-step explanation:

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.