Question

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

Answer 1

Answer:

Step-by-step explanation:

Property of tangent is " The tangent to the circle is perpendicular to the radius through the point of contact"

Given that the tangents are drawn at the ends of a diameter.

Let us assume that the two ends of diameter are A and B

Let the centre of circle be O

First, draw a tangent to circle at point A.

Thus, the tangent drawn at A is perpendicular to radius OA

Now, draw a tangent to circle at point B.

Thus, the tangent drawn at B is perpendicular to radius OB

Also, AB is a straight line as AB is a diameter

According to property of parallel lines, the interior and exterior angle are equal

Here the interior and exterior angle is equal to 90. (as tangent is perpendicular to radius)

Thus, the tangents drawn at the ends of a diameter of circle 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.