Q. 104.0( 24 Votes )

Prove that a diameter AB of a circle bisects all those chords which are parallel to the tangent at the point A.

Let AB be diameter in a circle with center O and MN is the chord at point A .

And CD be any chord parallel to MN intersecting AB at E

To Prove : AB bisects CD .

Proof :

OA MN [Tangent at a point on the circle is perpendicular to the radius through point of contact ]

MAO = 90°

Also, CEO = 90°

[ if E point lies on the OA then by Corresponding angles and if E lies on OB then by sum of co-interior angles ]

So, we have OE CD

CE = ED [ Perpendicular drawn from the center of a circle to chord bisects the chord ]

Implies that AB bisects CD

Hence Proved !

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