Question

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

Answer 1

Step 1: Identify what the given 22° angle represents

The given angle is at point A, between chords

A–B (a diameter), and

A–C.

∠BAC = 22°

By the Inscribed Angle Theorem, an inscribed angle equals half the measure of the arc it intercepts:

∠BAC = `1/2`​ BC

`hat(BC)` = 2 × 22° = 44°

Step 2: Determine the arc intercepted by angle x at D

Angle x = ∠ADC is an inscribed angle that intercepts arc AC, but crucially:

It must be the arc not containing D.

D is on the upper part of the circle.

So the intercepted arc AC is the lower arc, which passes through B.

Arc AB = 180° (since AB is a diameter).

Arc BC = 44° (found above).

`hat(AC)` = 180° + 44° = 224°

Step 3: Use the Inscribed Angle Theorem

∠ADC = `1/2` ​AC

x = `1/2` × 224°

x = 112°