Maths In Rugby Research Paper

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Maths in Rugby Introduction
This investigation focuses on the Maths related to scoring conversions in rugby. Having played rugby for many years I still find it tough to score conversions under pressure. Consequently the aim of this investigation is to develop a model that proposes optimal positions on the rugby pitch to take the conversion at along different vertical transect of the field.
In order to score a conversion, a try must be scored first. A try is scored by placing the ball on the ground, while in hand, at any point in the pitch behind the try-line but before the dead ball line. The referee then marks the point where the try has been scored and the conversion can then be taken at any point along the line running through the
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Below is an example equation using hypothetical values: Let x = 15 meters Let d= 7 meters (standard width of goal posts in Rugby Union) y= √(〖15〗^2-(7/2)^2 ) y=√(225-〖(3.5)〗^2 ) y=√212.75 y=14.58
The optimal position to take the conversion 15 meters away from the centre of the posts is 14.58 meters from the try line, along the conversion line.

Hughes’s Model
After doing some research I came across Hughes’s Model. It is an extension of my model as it provides points along a conversion line that allow the kicker the best chance to successfully convert the ball.

In order to find the position that provides the maximum angle, a series of expanding circles with AB as a chord can be drawn. All the inscribed angles in each of the circles are equal (inscribed angles on the same chord law) and these sets of equal angles decrease with an increase in size of the circle (exterior angle theorem for a triangle). For example, angle AFB = angle ACB and angle ADB>AGB. Consequently, the optimal position would be found where the conversion line forms a tangent with one of the circles as it provides the largest angle along the conversion line (line TF) than any other point. This is shown in Figure 4
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Let points C, D and E be 3 points chosen along the conversion line TF. Using the exterior angle theorem it follows that angle ACB = AC1B = AC2B is always larger than angle ADB or AEB, or any angle formed by points A,B and any point outside of the circle for that matter. Therefore, the optimal position for kicking a conversion is found at a point along a circle that has the conversion line as its tangent.
That is Hughes’s model and it accurately provides optimal positions for kicking along certain conversion lines. However, this model is limited as it would have no practical use during rugby matches as the position of these circles on the rugby field would be impossible to remember. Consequently, I have decided to look at the loci of the optimal point C as point T (grounding point), and consequently the conversion line moves along the try-line.
In order to determine the loci of point C as point T moves along the try-line I used a computer software called Sketchpad. The ‘trace locus’ function allowed me to trace the loci of point C as it moved as the conversion line moved along the try-line in the circles shown in Figure 4. These 4 loci were then joined up with a line of best fit and produced the diagram shown in Figure 6
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