Alternate Methods

Have a go at the angle problem below.

The solution is \(x=38\)

Based on what I've seen, I suspect that most people will start to work towards the centre in one of two ways:

1) Find \(\angle OAC\) and \(\angle CBO\) since the radii and tangents are perpendicular, then draw a new line OC to form two isosceles triangles. Use those base angles to find the angle at the centre and then use angles in a kite.

2) Draw a line segment AB and use the Alternate Segment Theorem to find \(\angle CBA\) and \(\angle BAC\). Use those angles to find \(\angle PAB\) and \(\angle ABP\), then use the angles in triangle \(APB\) to find \(x\). Some might even use symmetry to cut down on some work.

However, out of the hundreds of students that I've seen attempt this question, only two have ever worked outwards. Maybe this isn't surprising, given that we want to find \(x\) so we instinctively head in that direction, but it's interesting to consider there might be more efficient methods by taking a less direct approach.

 Did you and your students work inwards or outwards?