Week 3

Identify and apply circle definitions and properties, including: centre, radius, chord, diameter, circumference, tangent, arc, sector, segment.

This is a rather mundane list, though I have to admit that when it comes to 'sector' and 'segment', I still can't say with confidence which is which - though I have learnt that for me 'segment' is definitely not like the segment of an orange.

This week's tasks do not really test students' knowledge of the various terms in the list - I'm assuming they know what the terms mean. Instead, the tasks try to promote a better feel for some of the terms, but without challenging students too deeply. 

It is in Weeks 4, 5, 6, 7 and 8 that we earnestly apply circle properties This later content is based on a heading intended only for the 'more highly attaining students', though I hope it will have much wider appeal.

MONDAY: Here we are effectively being asked to compare the shapes of two segments. Are they similar?

It is quite challenging to compare curved shapes. How, for example, do we compare the 'angles' at, say, A and C? Intuitively, the 'corner' at A seems less sharp than that at C. We can look at this more analytically by thinking of the angles that the tangents at A and C make with their respective chords - do some students approach the task in this way?

In the diagram below we have enlarged the segment formed by the arc CD and chord CD, using a scale factor ×2. As we can see, the result is not congruent to the segment formed by the arc AB and chord AB.

Here's a perhaps more grounded approach (below). M is the midpoint of chord AB, so Q is the midpoint of arc AB and chord AP = chord CD. As we can see, the length of arc AP is less than the length of arc AQ.

Here are four interesting solutions posted on Twitter. The first two have similarities to the grounded approach, above. The third considers a special (and extreme) case, while the fourth uses a powerful, but far from grounded (!), argument.

TUESDAY: Here we do have an enlargement, with a scale factor ×2 and with the centre at the centre of the two circles. So the two arcs are similar, with one twice the length of the other.

WEDNESDAY: Here we are comparing an arc (minor arc, that is) with the chord joining its end-points. It is clear that the arc's length is always greater than that of the chord, but does the relationship change as the arc's length changes?

 

As the arc length gets smaller the arc and chord get closer in length: more precisely, the ratio of the lengths approaches 1. How readily do students appreciate this? As the arc length increases the ratio of arc-length to chord-length increases. Its maximum occurs when the (minor) chord is a semicircle, when of course its length is 𝜋r and the chord's length is 2r. The ratio 𝜋r/2r is about 1.5 which is less than 2.

 

THURSDAY: Here we are asked to compare the area of the non-overlapping parts of a sector and a segment of a circle.

When P is nearer to O than to C, as in the given diagram, it is fairly obvious that the area of V (the pure yellow region) is greater than that of U (the pure green region). But what happens when P is at the midpoint of OC? We can think of the two regions as two 'triangles', with the same base and with the same height. However, the yellow 'triangle' V has a convex side, so its area is greater than that of the green triangle U. As P moves closer to C, the area of the yellow region will decrease while that of the green region increases; at some instant the areas will be equal.

FRIDAY: We can use the diagram in this task to confirm some of our thinking in the previous task: here, point P is at the midpoint of OC and it is fairly clear that yellow region is larger than the green region. However, we now have to consider a third, brown, region. Is this smaller or larger than the yellow region?

It is quite difficult to compare the areas of the yellow and brown regions on a 'purely' visual basis, and it is likely that students will be divided as to which is larger. A useful, more analytical, approach is to divide the regions into triangles - moreover, triangles which are congruent, as in the diagram below. It is now fairly clear that the dark brown region is greater than the dark yellow region, so the brown region as a whole is greater than the yellow region as a whole.

If the radius of the circle is 2 units, then OP = 1 unit. Note also that angle BOP = 60˚. (Why?)

So the area (in unit squares) of triangle BOP = ½ × 1 × √3 = ½√3,
the area of triangle EOC = 4 × area of BOP = 2√3,
and the area of sector BOC = ⅙ of 𝜋2² = ⅔𝜋;

In turn, the area of the green region (G) = ½√3,
the area of the yellow region (Y) = ⅔𝜋 – ½√3,
and the area of the brown region (B) = 2√3 – ⅔𝜋.

In turn, Y – G = ⅔𝜋 – √3, which is about 2 – 1.7 which is greater than 1,
and B – Y = 2½√3 – ⁴⁄₃𝜋, which is about 4.33 – 4.19, which is (slightly) greater than 1.