Name: Tangents From an Exterior Point: Illustration
Uploaded: 2019-07-19
Duration: 4 min 47 s
Description: If you lift a circular object with a pair of tongs, then the tongs are two tangents to the circular object. Let us study the properties of tangents drawn fro...

Question

Prove that the tangent drawn at the mid-point of an arc of a circle is parallel to the chord joining the end points of the arc.

Accepted Answer

In the given figure, $C$ is the mid point of the minor arc $AB$ of the circle with centre $O$ .

$P Q$ is the tangent to the given circle through point $C$ .

To Prove: Tangent drawn at the mid point of the arc $\overset{︹}{A B}$ of a circle is parallel to the chord joining the end point of the arc $\overset{︹}{A B}$ , i.e $A B | | P Q .$

Proof: $C$ is the mid point of the minor arc $AB$

$\Rightarrow minor arc AC = minor arc BC$

$\Rightarrow A C = B C$ .

So, $∆ A B C$ is an isosceles triangle.

Therefore, the perpendicular bisector of side AB of $∆ A B C$ passes through the vertex $C$ .

We know that the perpendicular bisector of a chord passes through the centre of the circle.

Since $AB$ is a chord of the circle so, the perpendicular bisector of AB passes through the centre $O$ .

Thus, it is clear that the perpendicular bisector of $AB$ passes through the points $O$ and $C$ .

Therefore, $A B ⊥ O C$

Now, PQ is the tangent to the circle through the point C on the circle.

Therefore, $P Q ⊥ O C$ [Tangent to a circle is perpendicular to its radius through the point of contact].

Prove that the tangent drawn at the mid-point of an arc of a circle is parallel to the chord joining the end points of the arc.

Important Questions on Circles

Learn and Prepare for any exam you want !