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.

Answer 1

Given,C is the mid-point of the minor arc PQ and O is the centre of the circle and AB is tangent to the circle through point C.

Construction: Draw PC and QC.

To Prove: $PQ\parallel AB$

Proof:It is given that C is the mid-point of the arc PQ.

So, Minor arc PC=Minor arc QC

$\Rightarrow PC=QC$

Hence $△PQC$ is an isosceles triangle.

Thus the perpendicular bisector of the side PQ of $△PQC$ passes through vertex C.

But we know that the perpendicular bisector of chord passes through centre of the circle.

So,the perpendicular bisector of PQ passes through centre O of the circle.

Thus,the perpendicular bisector of PQ passes through points O and C.

$\Rightarrow PQ\perp OC....\left(i\right)$

AB is perpendicular to the circle through the point C on the circle.

$\Rightarrow AB\perp OC....\left(ii\right)$ (Radius of a circle is perpendicular to the tangent through the point of contact.)


From equations (i)and (ii),the chord PQ and tangent AB of the circle are perpendicular to the same line OC.

Hence,$AB\parallel PQ$

or $PQ\parallel AB$ Hence proved.