In a circle with centre

MATHEMATICS · Multi-part · CBSE · Mathematics · Circles · Circles

Problem

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}

. (i) Prove that

x = 8

. (ii) Hence calculate the measures of

\angle AOB

\angle ACB

and

\angle ADB

. (iii) Determine the reflex angle

\angle AOB

. (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