Question

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

Answer 1

Hint: There are three terms in the question, diameter, chords and tangent. We know that diameter is the longest chord. The tangent is perpendicular to the circle at an angle. So, we need to prove that the chord that is parallel to the tangent at a given point on the circle, it bisects the diameter. We shall do so by drawing the circle, then the tangents, chords and diameter.

Complete step-by-step answer:
Let us draw the below given figure as per the question.

Let ‘I’ be the tangent to the circle at ‘B’. ‘AB’ is the diameter that passes through the chord ‘CD’ at the point ‘P’. To prove that the diameter ‘AB’ bisects the chord, we need to prove that $CP = DP$.
We know that ‘I’ is the tangent. Therefore,
$AB \bot I$
Also, from the above figure, we can say that ,
$I\parallel CD$
CD is the chord of the circle,
$ \Rightarrow OP \bot CD$
We know that the perpendicular drawn from the centre of the chord, bisects the chord.
Therefore, diameter AB bisects the chord CD,
$ \Rightarrow CP = DP$

Note: Read the terms properly and draw a suitable figure to prove the given. Make sure that the tangent and the chord are parallel to each other. Draw the diameter first, draw a chord perpendicular to the diameter and a tangent parallel to the chord. Only after drawing, we can prove, so make sure that the figure is drawn as per the question.