Mathematics

Problem

In a circle with centre

O

, points

A

and

B

are fixed on the circumference. The chord

AB

subtends an angle of

8x^{\circ}

at the centre. A point

C

is chosen on the minor arc

AB

so that

\angle ACB = (3x + 8)^{\circ}

, and a point

D

is chosen on the major arc

AB

so that

\angle ADB = (10x + 68)^{\circ}

. a. (i) Prove that

x = 8

. b. (ii) Hence calculate the measures of

\angle AOB

\angle ACB

and

\angle ADB

. c. (iii) Determine the reflex angle

\angle AOB

. d. (iv) Verify that

\angle ACB + \angle ADB = 180^{\circ}

and explain why this result holds for any choice of points

C

and

D

on the two arcs of

AB

Intuitive explanation

The inscribed angle theorem is your key tool here: an inscribed angle is always half the central angle that subtends the same arc. Since $C$ and $D$ lie on opposite arcs, they "see" complementary portions of the circle, which means their inscribed angles relate to the central angle $8x°$ in predictable ways. Set up equations using this relationship, then solve for $x$ to find the configuration.

Intuitive explanation

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