Q18 of 62 Page 10

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.

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]


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!


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