Question

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

Answer 1

✒ GiVen :

• OA ⊥ PQ
• OB ⊥ RS

✒ To Prove :

• The tangents drawn at the ends of a diameter of a circle are parallel.

✒ SoluTion :

• Here, the tangents PQ and RS are drawn at points A and B, respectively.

Let AB be the diameter of a circle, with centre O respectively,

We know that,

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

So,

∴ OA ⊥ PQ and OB ⊥ RS

$\implies$ ∠OBR = 90°

$\implies$ ∠OAQ = 90°

$\implies$ ∠OAP = 90°

$\implies$ ∠OBS = 90°

Also, from the figure,

$\implies$ ∠OBR = ∠OAQ

$\implies$ ∠OBS = ∠OAP

Hence, these are the pair of alternate interior angles.

⛈ Since alternate interior angles are equal, the lines PQ and RS are parallel to each other. ⛈

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

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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