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.

Answer 1

Let us draw a circle in which AMB is an arc and M is the mid-point 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 mid point 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]

Now, ∠BMT' = ∠BAM  ...[Angle between tangent and the chord equals angle made by the chord in alternate segment] [2]

From [1] and [2]

∠ABM = ∠BMT'

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

Hence Proved!