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Mathematics-Online problems:

Problem of the week

The figure shows the path a billiard ball travels that returns to its starting point. This closed polygon is called a five-cushion.

\includegraphics[width=0.6\linewidth]{TdM_08_A4_bild1.eps}

Determine the coordinates of the five reflection points marked by the small circles. Also construct a three-cushion and a four-cushion starting at the same point and bouncing on the cushions $ \bigcirc\hspace*{-2.8mm}3$ $ \bigcirc\hspace*{-2.8mm}2$ $ \bigcirc\hspace*{-2.8mm}1$ and $ \bigcirc\hspace*{-2.8mm}2$ $ \bigcirc\hspace*{-2.8mm}3$ $ \bigcirc\hspace*{-2.8mm}4$ $ \bigcirc\hspace*{-2.8mm}1$ , respectively.


Answer:

Five-cushion:

$ \Big( a,$ $ b \Big)$, $ \Big( $ $ a, b \Big)$, $ \Big( 0,$ $ b \Big)$, $ \Big( a,$ $ b \Big)$, $ \Big( $ $ a, 0 \Big)$

Three-cushion:

$ \Big( $ $ a, b \Big)$, $ \Big( a,$ $ b \Big)$, $ \Big( $ $ a, 0 \Big)$

Four-cushion:

$ \Big( a,$ $ b \Big)$, $ \Big( $ $ a, b \Big)$, $ \Big( 0,$ $ b \Big)$, $ \Big( $ $ a, 0 \Big)$

(The results should be correct to four decimal places.)


   

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