Question

In the given figure, OD is perpendicular to the chord AB of a circle whose center is O. If BC is a diameter, show that CA = 2 OD.

Answer 1

Since OD ⊥ AB and the perpendicular drawn from the centre to a chord bisects the chord.
∴ D is the midpoint of AB.
Also, O being the center is the midpoint of BC.
Thus, in Δ ABC, D and O are the midpoint of AB and BC respectively.
Therefore, OD || AC
and OD = `1/2`CA            ....[ ∵ Segment joining the midpoints of two sides of a triangle is half of the third side ]
⇒ CA = 2OD                    ...Hence proved.