Question

OD is perpendicular to chord AB of a circle whose centre is O. If BC is a diameter, prove
that CA = 2 (OD)​

Answer 1

Perpendicular from the centre of a circle to a chord bisects the chord

We know that OB⊥AB

From the figure we know that D is the midpoint of AB

We get

AD=BD

We also know that O is the midpoint of BC

We get

OC=OB

Consider △ABC

Using the midpoint theorem

We get OB∥AC and

OD= 2/1 ×AC

By cross multiplication

AC=2×OD

Therefore, it is proved that AC∥DO and AC=2×OD

Answer 2

Answer:

Perpendicular from the centre of a circle to a chord bisects the chord

We know that OB⊥AB

From the figure we know that D is the midpoint of AB

We get

AD=BD

We also know that O is the midpoint of BC

We get

OC=OB

Consider △ABC

Using the midpoint theorem

We get OB∥AC and

OD=

2

1

×AC

By cross multiplication

AC=2×OD

Therefore, it is proved that AC∥DO and AC=2×OD