Question

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

Answer 1

In the diagram, the 40° angle at A is between AB and AD, not between AB and AC. So it is ∠BAD, not ∠BAC.

Therefore, ∠BAD = 40° subtends arc BD, not arc BC.

So,

arc BD = 2 × 40° = 80°.

The 120° angle at P is formed by the intersection of chords AD and BC. For two chords intersecting inside a circle,

angle between them is `1/2` (sum of the intercepted arcs).

∠APC intercepts arcs AC and BD, so

120° = `1/2`(arc AC + arc BD)

240° = arc AC + 80°

arc AC = 160°.

Angle x at D is ∠ADC, an inscribed angle subtending arc AC:

x = `1/2` (arc AC) = `1/2` × 160° = 80°.

So the correct value is x = 80°, and the reasoning in your screenshot (using ∠BAC and arc BC) is based on misreading which chords form the 40° angle.