(a) Prove that the angle which an arc of a circle subtends at the centre is twice that which it subtends at any point on the remaining part of the circumfer...

Answer Details

(a) Theorem: the angle an arc subtends at the centre is twice the angle it subtends at the circumference.

Let \(O\) be the centre, and let arc \(AB\) subtend \(\angle AOB\) at the centre and \(\angle ACB\) at a point \(C\) on the remaining circumference. Join \(CO\) and produce it to \(X\).

Triangle \(OAC\) is isosceles because \(OA = OC\) (radii), so \(\angle OAC = \angle OCA\). The exterior angle equals the sum of the interior opposite angles:

\[\angle AOX = \angle OAC + \angle OCA = 2\,\angle OCA.\]

Likewise, from isosceles triangle \(OBC\), \(\angle BOX = 2\,\angle OCB\). Adding,

\[\angle AOB = \angle AOX + \angle BOX = 2(\angle OCA + \angle OCB) = 2\,\angle ACB.\]

Hence the central angle is twice the inscribed angle on the same arc. (Q.E.D.)

(b) \(O\) is the centre of circle \(ACDB\), with \(\angle CAO = 26^\circ\) and \(\angle AOB = 130^\circ\); \(AD\) is the diameter through \(O\) (the dotted line \(A\!-\!O\!-\!D\)).

First find \(\angle AOC\). In triangle \(OAC\), \(OA = OC\) (radii), so it is isosceles with base angles equal:

\[\angle OCA = \angle OAC = 26^\circ \;\Rightarrow\; \angle AOC = 180^\circ - 26^\circ - 26^\circ = 128^\circ.\]

Since \(A,O,D\) are collinear (diameter), angles on the straight line at \(O\) give

\[\angle COD = 180^\circ - \angle AOC = 180^\circ - 128^\circ = 52^\circ,\qquad \angle BOD = 180^\circ - \angle AOB = 180^\circ - 130^\circ = 50^\circ.\]

(ii) \(\angle COB\). With \(C\) above and \(B\) below the diameter, \(\angle COB\) is made up of \(\angle COD\) and \(\angle DOB\):

\[\angle COB = \angle COD + \angle BOD = 52^\circ + 50^\circ = \boxed{102^\circ}.\]

(i) \(\angle OBC\). Triangle \(OBC\) is isosceles since \(OB = OC\) (radii), so its base angles are equal:

\[\angle OBC = \angle OCB = \frac{180^\circ - \angle COB}{2} = \frac{180^\circ - 102^\circ}{2} = \frac{78^\circ}{2} = \boxed{39^\circ}.\]