In Fig. 5, OD is perpendicular to chord AB of a circle whose centre is O. If BC is a diameter prove that CA = 2 OD.

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Solution

This again is application of mid-point theorem.
You know that any line perpendicular from centre of a circle to a chord, bisects the chord.
Hence, OD bisects AB or D is midpoint of AB.
O is midpoint of diameter BC.
From midpoint theorem, OD is parallel to AC and equal to 1/2 AC.
In other words, AC = 2 OD .... Proved.

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