The radius

The radius of a circle is a line, which starts at the centre of the circle and ends on a point on the circumference. The radius is half the size of the diameter.

An isosceles triangle within a circle, the legs equal to the radius of the circle

Looking at the diagram above, since OA and OB are radii of the circle, then OA = OB, therefore\(AOB\) forms an isosceles triangle inside the circle.

Also, note that since \(\Delta AOB\) is isosceles, then \(\angle OAB = \angle OBA\), this means that the angles at A and B are equal.

Question

In this circle, calculate the size of \(\angle OBA\).

An isosceles triangle within a circle and a 136∞ angle between the diameter and the radius

\(\angle AOC\) is a straight angle, so will add up to \(180^\circ\). This is the straight line \(AC\).

\(\angle AOB = 180^\circ - 136^\circ = 44^\circ\). This is the other angle at \(O\).

\(\Delta AOB\) is isosceles, so \(\angle OAB = \angle OBA\) and angles in a triangle will add up to \(180^\circ\).

This means the angles at \(A\) and \(B\) will have \(180^\circ- 44^\circ\) split between them equally.

\[\angle OBA = (180^\circ - 44^\circ ) \div 2 = 68^\circ\]