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.

At A, ∠CAB = 18°.

This is an inscribed angle standing on arc CB, so

arc CB = 2 × 18° = 36°.

Chords AC and BD intersect at E, and ∠AEB = 90°.

For two chords intersecting inside a circle:

∠AEB = `1/2` arc AB + arc CD.

So 90° = `1/2` arc AB + arc CD ⇒ arc AB + arc CD = 180°.

arc AB + arc BC + arc CD + arc DA = 360°.

Substitute arc BC = 36° and arc AB + arc CD = 180°:

180° + 36° + arc DA = 360° ⇒ arc DA = 144°.

Angle x = ∠DCA is an inscribed angle standing on arc DA, so 

x = `1/2` × 144° = 72°.