Optimum kicker position of the rugby ball to convert a try

In rugby you score a try when you move the ball into the opposition in-goal area and "ground" the ball. That is the only place a try is scored. You get 5 points for a try. When a try has been scored, two (2) additional points can be scored by kicking the ball so that it goes both over the cross bar and between the posts. This is called ‘converting the try’. The best kicker in the team uses a place kick for the conversion. If the kick is successful the try is "converted" and your team scores the two extra points. The kick is taken in line with where the try was scored. To make the angle of the kick easier, you can kick from as far away from the goal-line as you like, but the further you go away from the try line , the longer the kick. Scoring a try near the posts makes it easier to get the extra points for converting. Even after crossing the goal-line you can run towards the posts to get as near as you can to the posts before grounding the ball. The nearer the goal posts the better. The task You are a sports scientist that has been hired by a top rugby team. Your task is to investigate what the best distance is to place the ball, after a try, to increase the chances of getting the extra two points from a conversion, based on the position of where the try was scored. You will have to write a report for the coach, who won’t believe you without proof, of your findings.
You will use Geogebra to help you investigate this. You will need to find a rule that connects the distance from the goal post (x), where the try was “grounded”, and the distance from the goal line(y) that will give you the biggest angle (Theta). In the Geogebra you have a slider (a) that allows you to change the distance from the goal posts (x). You can then move point A until you get a maximum angle, Theta. What happens to the position of the tangent AF when (Theta) is at its maximum value? Was this always the case?   Proof You can use circle theorems to show that this is the best position regardless of the distance from the post. 1. Show that the angle AEB= angle AOM. What circle theorem did you use? 2. Show that angle ACB=180-angle AEB. What circle theorem did you use? 3. Show that angle ACD=angle AEB. What geometric fact did you use to do this? 4. Now by considering triangles Δ AOM and Δ ACD, show that angle DAC=angle MAO. What geometric fact did you use to prove this? 5. What is the size of angle FAO? What circle theorem gives you the answer? 6. Using your answer to 5, write an expression for the sum of the four angles that make  FAO. 7. You have shown that angle DAC=angle MAO, let angle DAC=angle MAO= phi and angle CAB=theta. Substitute those into your expression in 6. 8. Now rearrange the expression to make theta the subject. What can you say about angle FAD in order to maximize the size of angle theta? 9. Write a conclusion for your results using the therefore symbol. Now that you have shown that the maximum angle is achieved when the tangent line AF is perpendicular to the goal line, you can find an expression (a rule) for the best distance (y) in terms of the distance from the post (x). 1. Given that angle AFG is 900, show that AO and FG are parallel. (What theorem did you use?) 2. Show that CG=GB=2.8m, by explaining why point G bisects the line BC. (Hint: chords) 3. Write an expression for FG in terms of x. 4. Explain why FG=OB. 5. Explain why AF=OG=y. 6. You now have an expression for all the sides in ΔOGB in terms of x and y, use Pythagoras to find a rule for y in terms of x.