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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.


Solution

In this figure, C is the mid point of the minor arc 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 arc \(\widehat{AB}\) 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 arc AB.
minor arc AC = minor arc BC
AC = BC
This shows that ΔABC is an isosceles triangle.
Thus, the perpendicular bisector of the side AB of ΔABC passes through 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.
 ABOC
Now, PQ is the tangent to the corcle through the point C on the circle.
 PQOC [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.
 AB||PQ

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