The Mathematics of Everyday Life by Alfred S. Posamentier
Author:Alfred S. Posamentier
Language: eng
Format: epub, mobi, azw3
ISBN: 9781633883888
Publisher: Prometheus Books
Figure 5.16.
We will now reflect the target ball in the side cushion, which is geometrically equivalent to seeing the reflection of the target ball in the mirror that was placed against the side cushion. We then aim the ball at the mirror-reflection of the target ball with the straight-line segment shown in figure 5.17. The ball will then ricochet off the cushion at the point P and hit the target ball as desired.
Figure 5.17.
We can easily show that this is the shortest distance from the cue ball to the cushion to the target ball. This can be done by choosing any other point Q on the cushion, and comparing the distance traveled from the cue ball's starting point to the point Q and then to the target ball with the distance of the ball's previous path through point P. (See fig. 5.18.) The path from point C to P to T, or, as stated geometrically, CP + PT is equal to CP + PR, since triangle TPR is isosceles. Similarly, CQ + QT = CQ + QR. However, we know that the shortest distance between two points is a straight line, therefore, CP + PR < CQ + QR. Or, stated verbally, CP + PR is shorter than any other path from the cue ball to the cushion to the target ball.
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