DOCUMENTACIÓN REQUERIDA POR LAS FUENTES DE FINANCIAMIENTO 1. Introducción

The diagram in Figure 3.4f shows a statue on a column. Find the position at which the statue subtends the great- est angle at the person’s eye.

Comment

No lengths are given, and there are no obvious circles around.You could use some specific values. For example, the height of the column might be

6 m and the height of the statue 2 m. The person’s eye might be 1.6 m above the ground. The aim is to think geometrically rather than arithmetically.

Think of the statue as the chord of a circle and the angle subtended at the viewing point as an angle at the circumference.What circle properties come to mind triggered by that language?

This problem can be solved algebraically, of course, but this was not available to Regiomontanus! Moreover, it is often the case that a purely geometric solution is simpler and more elegant. As you have seen before, it is easy to construct a circle with the statue as chord but what will ensure that the angle subtended at the viewpoint is a maximum? Imagine the viewpoint coming closer and going farther away, and how the subtended angle changes (see Figure 3.4g). Consider the diagram on the right, which shows a circle with the statue as chord and passing through the viewpoint. Why does that viewpoint not give the maximum angle?

From these observations you can conclude that the maximum subtended angle will be when the horizontal line through the viewing point is a tangent to the circle with the statue as chord. So the problem now becomes finding a method to construct this circle. More specifically, how can the position of the centre be determined? The circle must pass through the top and bottom of the statue. Consider the invariant properties of the situation.What do you know about a chord in relation to the centre of a circle? From this you can construct a line that the centre must lie on. Now focus on the viewing point (the person’s eye). Because the line through the viewing point must be a tangent to the circle, the height of the centre above this line must be the radius of the circle. So now the radius is known and hence the centre can be found using one of the points, say the top, of the statue. A dynamic geometry software construction helps in verifying that no other position on the ground has a larger angle subtended by the statue. (See the interactive file ‘3f Regiomontanus’.)

3.5 PEDAGOGIC PERSPECTIVES

Because of the many invariant relationships connected with circles, they can be useful when reasoning, and they are involved in constructions that have nothing ostensibly to do with circles.

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Figure 3.4g

Reflection 3.4

What is the same and what is different about the reasoning involved in Tasks 3.4.1 and 3.4.2? And in Tasks 3.4.3 and 3.4.4?

Inner and Outer Tasks

The tasks offered in these first three chapters have revealed a large number of geo- metrical facts or theorems.The word theorem is of Greek origin and means, literally, a ‘seeing’.That has been their overt or outer purpose. But the tasks were chosen in order to afford you experience of different aspects of geometrical thinking and, in this chapter particular, geometrical reasoning. This was part of the inner purpose of the tasks, which could not be stated explicitly because then you would have been attend- ing to that and not gaining the experience. By engaging in the explicit outer task, you will have had opportunities to discern details of figures, to recognise relationships, to perceive properties (independent of the particular figure) and to develop your reason- ing on the basis of properties.You may not always have been aware that this was going on, but over a period of time working through this book you will be building layer upon layer of experience of subtle but important aspects of geometrical thinking.

Two Pedagogic Strategies: Say What You See and Same and Different

When pondering a figure, you will have noticed that a great deal rides on noticing elements and relationships that can be developed into a chain of reasoning. One of the best ways to help yourself, or a group of learners, to discern relevant and useful elements, sub-figures and relationships is literally to ‘say what you see’. Each person can say something, and each element pointed out will direct attention.You soon dis- cover that what is salient for one person is overlooked or ignored by another. Developing geometric thinking depends on learning to discern and to recognise geo- metric relationships that are fruitful, and this is best developed through participating in a group and hearing what others have to say.

You will have noticed the occasional use of the question ‘what is the same, and what is different about … ?’.This turns out to be an extremely fruitful type of ques- tion when there is more than one learner pondering two or more figures or sub-figures. What is the same and what different about two triangles can lead to seeing that they are congruent or similar; about two angles can lead to seeing that they are both subtended from the same chord of a circle, and so on.

The Role of Imagination

Powerful as dynamic geometry software is for revealing relationships, the most power- ful tool is your own imagination. Whenever possible it is wise to try to imagine a figure before sketching it, to sketch it before using dynamic geometry software, and when using dynamic geometry software, to make a conjecture as to what will happen before actually doing it (dragging, constructing a new element), thus creating an expectation. Expectation is a product of imagining, and without an expectation there is unlikely to be any surprise, yet geometry is a domain full of surprising relationships. Several times in tasks calling for a construction, it has been useful to imagine the construction as if it were completed. This introduces extra elements into a figure, which in turn can help you to recognise relationships, perceive these as properties and so, support your reasoning on the basis of properties.

Unexpected Invariants

In Task 3.2.5, it was shown that angles subtended on the same side of a chord in a circle must be equal. With care it is possible to define what you mean by ‘angle sub- tended by a chord in a circle’ so as to remove the need to specify ‘on the same side’. It is also the case that if you start with the property ‘all angles subtended (on the same side) of a segment are equal’, then the points satisfying that will lie on a circle with the segment as chord.

Many of the tasks in this chapter have attempted to highlight how the invariant relationships (perceived as properties independent of the particular figure) play an important role in the reasoning process when solving problems or designing con- structions. It quite often happens that textbook problems, and students’ attempted solutions, will unexpectedly raise interesting issues that when probed, reveal invari- ances. The following two tasks illustrate this. The first is ostensibly a very simple problem concerning circles.

The radii approach mentioned in the comment above divides the diagram into three congruent isosceles triangles and hence ∠AOB = ∠BOC = ∠COD = 180°/3.This forces the triangles to be equilateral, so in particular ∠AOB = ∠OBC. Since these are alternate angles, BC must be parallel to AD. Compare

this to the solution given by one student (Figure 3.5b). In Figure 3.5b the line BD is drawn.∠ADB = ∠DBC because they are subtended by equal chords (AB and CD). But since they are also alternate angles, BC must be parallel to AD.

It is certainly very different to the radii approach and

is striking in its simplicity and elegance.Apart from the final step concerning alternate angles, the concepts used are quite different too. In fact, it just uses the invariant prop- erty of equal angles subtended by equal chords in a circle. But what about the information used? Again, there is a striking difference here. The fact that AB = CD was crucial, but the fact that BC is also equal to these sides was ignored. This means that it must be redundant information as far as the parallelism of BC and AD is con- cerned. Not only that, but also the fact that AD is a diameter was ignored, so that too must be redundant.That is, the original diagram can now be seen as a special case of a more general invariance: if two equal chords AB and CD are drawn anywhere in a 54 BLOCK 1