Question

In this figure, C is the mid point of the minor are AB, O is the centre of the circle and PQ is the tangent to the circle through the point C.Prove that the tangent drawn at the mid point of the are $\widehat{AB}$ of a circle is parallel to the chord joining the end points of the arc

Answer 1

In this figure, C is the mid point of the minor are AB, O is the centre of the circle and PQ is the tangent to the circle through the point C.
To prove: the tangent drawn at the mid point of the are $\widehat{AB}$ of a circle is parallel to the chord joining the end points of the arc $\widehat{AB}$
I.e., it is required to prove that AB ||PQ.
It is given that C is the mid -point of the are AB.
$\therefore$ minor are AC =minor are BC
$\Rightarrow AC=BC$
This shows that $\Delta ABC$ is an isosceles triangle.
Thus, the perpendicular bisector of the side AB of $\Delta ABC$ passes throgh the vertex C.
It is also known that the perpendicular bisector of a chord passes through the centre of the circle.
Since AB is a chord to the circle, so, the perpendicular bisector of AB passes through the centre O of the circle.
Thus, it is clear that the perpendicular bisector of AB passes through points O and C
$\therefore AB\perp OC$
Now, PQ is the tangent to the circle through the point C on the circle.
$\therefore PQ\perp OC$ [Tangent to a circle is perpendicular to its radius through the point of contact]
Now, the chord AB and the tangent PQ of the circle are perpendicular to the same line segment OC.
$\therefore AB||PQ$