{Apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results} (3)
This is another set of tasks where we look at properties derived from the circle theorems. This time we focus on the angle at the centre theorem, as in the question below, which is from a 1959 GCE examination paper. Indeed, the tasks are all variations of this one, in which we are presented with two circles, one of which passes through the centre of the other.
MONDAY: This task provides an introduction to the week's scenario. It is perhaps more immediately accessible than later tasks in that it involves a specific angle - the 90˚ interior angle of a square. However, students still have work to do, to decide how to use this and other implicit angles in the given figure.
Angle ADC = 90˚, so angle ADB = 45˚, so angle AOB = 45˚, so angle APB = 22½˚.
Of course, one would need to justify each step.
Another approach is to work backwards, at least for a while, as here:
I could find the desired angle APB, if I knew angle AOB, which I could find if I knew angle ADB, which I can find because it is half of angle ADB which is 90˚.
TUESDAY: This is the original GCE question, but without the initial hint about the angle at the centre theorem.
This is a subtle and demanding task, even with the angle at the centre hint. But it has the potential of giving students a hugely important experience: of perhaps struggling with a task, but having a sudden insight that solves it!
We can solve the task if we can show that the triangle BQP (which is not shown explicitly in the diagram) is isosceles. But what do we know about its base angles? - ah, that their sum is equal to the exterior angle AQB. And we potentially know a lot about this angle as it is related to two other angles in the diagram (albeit angles that are also not shown explicitly!).
You might decide to scaffold the task for students, for example by drawing the line-segment BP, perhaps also AO and BO, although that would hugely reduce the insight that students could at some point gain and take pleasure from!
WEDNESDAY: This is essentially the same situation as before, except that Q has moved to a position on the smaller circle that takes it outside the larger circle and takes P inside the smaller circle. Can we use the same or similar arguments as before?
THURSDAY: Here Q has moved further along the smaller circle, leaving it still outside the larger circle but with P back outside the smaller circle. Can we still use our earlier arguments?
FRIDAY: Here Q has moved further still, until it coincides with point A and essentially disappears. How does this affect our earlier arguments?
