# Prove that the tangent drawn at the midpoint of an arc of a circle is parallel to the chorf joining the end points of the arc

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by bittu10

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by bittu10

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We have to show the tangent drawn at the mid-point of the arc PQ of a circle is parallel

to the chord joining the end points of the arc PQ.

We will show PQ ║AB.

It is given that C is the midpoint point of the arc PQ.

So, arc PC = arc CQ.

⇒PC = CQ

This shows that ΔPQC is an isosceles triangle.

Thus, the perpendicular bisector of the side PQ of ΔPQC passes through vertex C.

The perpendicular bisector of a chord passes through the centre of the circle.

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

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

⇒PQ ⊥OC

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

⇒AB ⊥ OC

The chord PQ and the tangent PQ of the circle are perpendicular to the same line OC.

∴PQ || AB.