. 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.

Explanation

Let mid-point an arc AMB be M and TMT' be the tangent to the circle.
Join AB, AM and MB.
Since, arc AM=arc MB
`rArr"Chord "AM="Chord "MB`
In `DeltaAMB`, AM=MB
`impliesangleMAB=angleMBA`
[equl sides corresponding to the equal angle].....(i)

Since, TMT' is a tangent line.
`:.angleAMT=angleMBA` [angles in alternate segments are equal]
`angleMAB` [from Eq. (i)
But `angleAMT and angleMAB` are alternate angles, which is possible only when
AB//TMT'
Hence, the tangent drawn at the mid-point an arc of a circle is parallel to the chord joining the end points of the arc. Hence proved.