Question -
Answer -
Let mid-point of an arc AMB be M and TMT’ be the tangent to the circle.
Join AB, AM and MB.
Since, arc AM = arc MB
=> Chord AM = Chord MB
In ∆AMB, AM = MB
=> ∠MAB = ∠MBA ……(i)
[equal sides corresponding to the equal angle]
Since, TMT’ is a tangent line.
∠AMT = ∠MBA
[angle in alternate segment are equal]
∠AMT = ∠MAB [from Eq. (i)]
But ∠AMT and ∠MAB are alternate angles, which is possible only when AB || TMT’
Hence, 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.
Hence proved