Question

Question 9
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.

Answer 1

Let mid point of an arc AMB be M and TMT' be the tangent to the circle.
Join AB and MB
Since, arc AM = arc MB
$\Rightarrow \angle MAB=\angle MBA$
[Equal sides corresponding to the equal angle] ….(i)

Since . TMT' is a tangent line.
$\therefore \angle AMT=\angle MBA$
[Angles in alternate segments are equal]
= $\angle MAB$ From eq.(i)
But $\angle$AMT and $\angle$MAB are alternate angles, which is possible only when AB $\parallel$ TMT'
Hence, the tangent drawn at the midpoint of an arc of a circle is parallel to the chord joining the endpoints of the arc.