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.

Given:

Let us draw a circle in which AMB is an arc and M is the midpoint of the arc AMB. Joined AM and MB. Also, TT' is a tangent at point M on the circle.

To Prove: AB || TT'

Proof:

As M is the midpoint of Arc AMB

Arc AM = Arc MB

AM = MB [As equal chords cuts equal arcs]

ABM = BAM [Angles opposite to equal sides are equal] [1]

As we know angle between tangent and the chord equals angle made by the chord in alternate segment,

So,

BMT' = BAM [2]

From [1] and [2]

ABM = BMT'

So, AB || TT' [two lines are parallel if the interior alternate angles are equal]

Hence Proved!