Question

Find the value of x in the following figure, where O is the centre of the circle: 

Answer 1

Given

O is the centre of the circle.

∠AOD = 130° (central angle).

We need to find x = ∠BCD, an inscribed angle.

Step 1: Arc subtended by central angle AOD

A central angle equals the measure of the arc it subtends:

Arc AD = 130°

Step 2: Determine the arc subtended by angle x

Angle x = ∠BCD is an inscribed angle which subtends arc BD.

So we need the measure of arc BD.

The entire circle is 360°.

Arc AD = 130°

Arc AB is a straight line through the centre (diameter-like), so:

∠AOB = 180°

Arc AB = 180°

Arc BD = Arc AB − Arc AD

Arc BD = 180° − 130° = 50°

Step 3: Inscribed angle theorem

An inscribed angle equals half the arc it intercepts:

x = `1/2` × Arc BD

x = `1/2` × 50° = 25°