Week 8

{Apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results} (5)

Here we take another look at the fundamental circle theorem Angle at the centre. But this time we use the approach that appears in Euclid's Elements and is based on isosceles triangles.

Here is a classic example of how the theorem is stated, along with diagrams used to prove it. This is from Durell, 1939, A New Geometry for Schools. Durell's proof includes a fourth diagram in which angle AOB is reflex.

The first two tasks give students an opportunity to think about the relative size of the angle subtended by an arc as the vertex of the angle moves from a point on the circumference towards the centre of the circle and onwards to the circumference again. The subsequent two tasks involve steps similar to ones in the standard Angle at the centre proof. Finally we take a dynamic look at diagrams similar to two key diagrams in the formal proof and consider what happens if we extend the diagrams.

MONDAY: In this task it is possible, for students who know the Angle at the centre theorem, to determine the size of the angle APB when P is at U, O or V. However, it is also interesting to give the task to students who don't know, or haven't recently been reminded of, the theorem. What are their intuitions? What qualitative responses do they give about the size of the angle as the position of P changes?

Some students might think that the size of angle APB doesn't change as P travels from U to V. Some others might think it gradually gets larger and larger (or smaller and smaller). How many students correctly sense that the angle gets bigger as P approaches O and gets smaller again as it approaches V?

It is interesting to look more closely at what happens when P is near O. The angle APB gets bigger as P approaches O, and we know from the Angle at the centre theorem that its size when P is at O is twice the size of when P is at U (ot V). However, surprisingly perhaps, it does not attain its maximum at O, but at a point T (below) slightly closer to V, where the circle that passes through A and B and is tangential to UV, touches UV. (Why?)

TUESDAY: Here we encounter the Week 6 scenario again, where one circle goes through the centre of another. So the journey taken by the point P is more complex. Students who know the Angle at the centre theorem, should be able to determine the size of angle APB when P is at U or V (35˚) and when it is on the smaller circle (70˚). However, it should also again be interesting to assess the intuition of students who don't know, or don't quite recall, the theorem.

The Angle at the centre theorem tells us that angle APB has a value of 35˚ when P is at U or V, and a value of 2×35˚ = 70˚ when P is at either point where UV intersects the smaller given circle. When P is between those latter points, the angle is somewhat larger than 70˚.

As before, it turns out that the maximum is reached at the point T (below), where the circle that passes through A and B and is tangential to UV, touches UV.

WEDNESDAY: Here we present the 'default' form of the diagram that is used for the classic proof of Angle at the centre, where A and B lie on either side of the line through O and P. The task gives students the opportunity to (re)discover one of the key steps in the proof, though here we use specific angles. It is worth pointing out that we have pre-empted another key step: that of joining P to O and then extending PO to N - this step is crucial as it allows us to use a property that derives from the fact that P lies on the circle, namely that P is the same distance from the centre as A and B.

This step involves several radii which, being equal in length, provide us with some isosceles triangles. Using these triangles, we can show that angle APO = 78˚÷2 and angle BPO = 22˚÷2, and so APB = (78˚+22˚)÷2 = 100˚÷2 = 50˚. Working at a more general level, we could use a similar argument to show that angle APB = angle AOB ÷ 2.

THURSDAY: This is essentially the same as Wednesday's task, except that P has moved to a position where the centre O of the circle is no longer inside triangle APB.

We can show that angle APB = (100˚–20˚)÷2 = 50˚, as before.

FRIDAY: Here we examine what happens as P continues to move around the circle and we look, in particular, at what happens when P reaches B and then goes beyond B. The ideas here are quite challenging but hopefully students will find them intriguing, even if some are left mystified! [I have written about this dynamic approach in Mathematics in School (2003, 32, 2): Angle at the centre: taking a point for a walk.]

These diagrams don't include the line PO produced but instead we have produced PB to a point X. The latter allows us to make sense of the situation when P is at B (otherwise, the angle b would disappear at that point). Here PX (or BX) is a tangent to the circle, so we could use (or derive) the Angle between tangent and chord theorem to show that b is still equal to 50˚. The fourth diagram evokes another theorem, about the internal and external angles of a cyclic quadrilateral, although we can confine ourselves to Angle at the centre (this time, an angle of 260˚) to show that b is still 50˚ (ie 180˚ – 260˚/2).