mathematical investigation project

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Introduction

The aim of our project is to investigate the number of chords and points of intersection within a circle with a variable number of points on the circumference and dissected from each point to every other point on the outside without any more than two lines intersecting each other (Fig. 1.1). Our secondary aim is to discover the interrelationship between these two.

( Fig. 1.1 )

For example, in Fig. 1.1, fourth in a series of circles starting with the first with one point on the circumference, the circle has four points on the circumference, six lines joining the points, and one point of intersection.

Chords

In the course of our project, we have found the number of chords in each of the circles with an increasing number of points on the outside. For Figures 2.1 through 2.6, there are 0, 1, 3, 6, 10 and 15 chords respectively. We observed that the number of chords increased proportionately to the number of points on the circumference of the circle. For example, in the third figure (Fig. 2.3), there are 3 chords and in the fourth figure (Fig. 2.4), there are 6 chords, a difference of 3. We have also found that this is true for all the figures. Thereafter, we realised that the formula for the number of chords is , where n is the number of the sequence in the figures.

Fig. 2.1 (0 chords) Fig. 2.2 (1 chord) Fig. 2.3 (3 chords)

Fig. 2.4 (6 chords) Fig. 2.5 (10 chords) Fig. 2.6 (15 chords) ( As it can be seen, the sequence of chords begin at the second figure instead of the first. )The normal formula for finding triangular numbers is . As in the case of this sequence, which starts at 0 chords in the first figure, n has to be decreased by 1 due to the first figure in the sequence. By subtracting n by 1, we can get the formula .

( Fig. 1 ) ( Fig. 2 )

Conclusion

In conclusion, we have discovered the formulae for finding both the number of chords and the number of points of intersection within a circle with a progressively increasing number of external points.

This project also shows that a relatively simple project like this can be used effectively in the real world, the most obvious of which are the planning of road layouts and architecture.

Therefore, we can proudly say that we think this project is a success in all ways given the limited amount of time that we had.

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