Question

Question 10
Prove that a diameter AB of a circle bisects all those chords which are parallel to the tangent at the point A.

Answer 1

Given AB is a diameter from point A. Draw a chord CD parallel to the tangent MAN.

So, CD is a chord of the circle and OA is a radius of the circle.
$\angle MAO={90}^{\circ }$
[ tangents at any point of a circle is perpendicular to the radius through the point of contact]
$\angle CEO=\angle MAO$
$\therefore \angle CEO={90}^{\circ }$
Thus, OE bisects CD, [ perpendicular from centre of circle to chord bisects the chord]
Similarly, the diameter AB bisects all chords which are parallel to the tangent at the point A.