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

9th Class Maths – NCERT Important Questions

Ans:

Step 1:

Let’s make a diagram for better understanding of the question:

Here PQ is the diameter of the circle with center O.

AB and CD are the two tangents, touching the circle, at points P and Q, respectively.

Step 2: 

Since PQ is the diameter with center O

∴ OP and OQ are radii of the circle

We know that a radius is perpendicular to tangent,

∴ when OP is radius and AB is tangent, then ∠ APO = ∠ BPO = 90⁰

and when OQ is radius and CD is tangent, then ∠ CQO = ∠ DQO = 90⁰

∴ ∠ APO = ∠ BPO = ∠ CQO = ∠ DQO = 90⁰

Step 3:

We know that when a line intersects two parallel lines, its alternate interior angles are equal.

Alternatively, if alternate interior angles are equal, two lines being intersected are parallel.

Here, we can see that PQ is intersecting 2 lines, AB and CD.

∠ BPO and ∠ CQO are alternate interior angles as well as equal ( = 90⁰… proved in Step 2)

∴ AB and CD are parallel lines.

Hence Proved !

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