In the adjoining figure, AB is a chord of a circle with centre O and BC is a diameter. If OD ⟂ AB, show that CA = 2OD and CA ∥ OD.
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Question

In the adjoining figure, AB is a chord of a circle with centre O and BC is a diameter. If OD ⟂ AB, show that CA = 2OD and CA ∥ OD.

KnowledgeBoat · Accepted Answer

Since, the perpendicular to a chord from the centre of the circle bisects the chord,

∴ AD = DB

We can say that D is mid-point of AB.

Since, BC is diameter and O is center so, OB = OC = radius.

We can say that O is mid-point of BC.

In △ABC,

Since, D is mid-point of AB and O is mid-point of BC.

By mid-point theorem,

⇒ OD || AC and OD = $\dfrac{1}{2}$ AC

⇒ AC = 2OD.

Hence, proved that CA = 2OD and CA ∥ OD.

Mathematics