GEOMETRY AND A GAME OF BILLIARDS
Maths Workshop
 
THREE CUSHIONS

Did you get any solutions?

1.- The first solution we get is the path: "bottom-top-bottom".

We find the reflections of A and B about the bottom cushion (A' and B') and the reflection of B' about the top cushion (B'''). The solution is the point where line A' B''' intersects the bottom cushion.

Sol: R = "35x-15y=275" "y=0" = (55/7,0) (7'857,0)

2.- Another solution we get is: "bottom-top-right".

Sol: R = "25x+27y=229" "y=0" = (229/25,0) (9'16,0)

3.- Another possible solution is "bottom-right-top".

Here: B' is the reflection of B about the top cushion and the reflection of B'' about the right-hand cushion.

R= the point where the line A' B'' intersects the bottom cushion.

However, with the information given in our example point B'' coincides with the previous example and this solution is not possible as the ball would first bounce off the top cushion before the right-hand one.

4.- And finally?: the path "bottom-right-left".

A'  = the reflection of A about the bottom cushion;
B'  = the reflection of B about the left-hand cushion;
B''' = the reflection of B' about the right-hand cushion;
R   = the point where line A' B''' intersects the bottom cushion.

Sol: R = "7x+71y=191" "y=0" = (191/7,0) (27'28,0)

5.- What about "bottom-left-top"?

We'll leave it up to you to find out.
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  Andrés Mateos Royo
 
Spanish Ministry of Education. Year 2001
 
 

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