Using Geometer's Sketchpad to model a pool shot

1.

First, we need to construct our virtual pool table:

(a)

Make a line segment of suitable size. It is recommended, though not necessary, that the segment be perfectly vertical or horizontal.

(b)

Select one of the endpoints and the line segment, then use ``construct -- perpendicular line."

(c)

Then select the perpendicular line, click construct ``point on perpendicular line."

(d)

Move the point to a suitable place.

(e)

Select the point and the first perpendicular line, then use the construct perpendicular line function again.

(f)

Now go to the other endpoint of the original line segment. Construct a perpendicular line there.

(g)

Construct the other corner by selecting the two intersecting lines and using ``construct -- point of intersection"

(h)

Make the top side of the ``table" into a segment by clicking on its endpoints and using ``construct -- segment"

(i)

Then construct the midpoint by using ``construct -- midpoint"

(j)

Do the same on the bottom side.

(k)

Now label the ``pockets" starting with the top left and going clockwise, A ... F.

2.

Now, put a point in an appropriate spot to use as your target ball, label it Y.

3.

As with most problems, there's more than one way to figure out where to shoot Y. Before we do anything weird, let's just try shooting Y at different spots on the rail and seeing what happens.

(a)

Select the segment AC and use ``construct -- point on segment"; this will be the point we shoot at.

(b)

Construct the segment YG.

(c)

Select AC and G and use ``construct -- perpindicular line". Once it's constructed, this line should still be selected; use ``transform -- mark mirror". We don't want this line cluttering up our picture, so select it and use ``display -- hide perpendicular line" to hide it.

(d)

Now, select YG and G, then use ``transform -- reflect" to reflect them about the line we just hid.

(e)

Select G then the new point you just created and ``construct -- ray". Then select just the new point and ``display -- hide point" to hide it.

(f)

Move G (if necessary) until the ray you just made intersects the bottom rail. Select the ray and FD then ``construct -- intersection". The intersection point should still be selected at this point, so use ``display -- label intersection" to label it; call it Z.

(g)

Remember that G is the point on the top rail that you're shooting at; as you move G around, Z shows where your bank shot will end up.

(h)

Let's measure some angles: select A, G, Y in that order, then use ``measure -- angle" to measure the angle. Next, select C, G, Z and again use ``measure -- angle".

(i)

What can you say about angles AGY and CGZ? What are AGY and CGZ when you move G to a spot where Z matches up with D (that is, when the bank shot will go in the pocket)?

(j)

What do you think of this method of solving the problem?

4.

Let's try a different method. First, move G over close to A so it won't get in the way (don't put G on top of A, or you may not be able to select it again). Next, we want to reflect the pocket D across the AC rail.

(a)

Select AC, use ``transform -- mark mirror."

(b)

Select D and CD, then use ``transform -- reflect"

(c)

Look up, find the new point. It should be already selected. Use ``display -- label point" to label the point D’. Note that D’ is the default name for this reflected point.

5.

Now, we just shoot our ball at the ``reflection" of D: construct the line segment between Y and D’.

6.

Select YD’ and AC. Click ``construct -- intersection."

7.

Label the intersection K.

8.

Now, we measure some angles.

(a)

Select the points A, K, Y in that order!!

(b)

Use ``measure -- angle." Note what happens if you click the points in the wrong order. This is the angle of incidence.

(c)

Now, measure the angle of refraction: construct the segment KD, and measure the angles DKY, D’KC. How does the angle of refraction compare to the angle of incidence?

9.

Try moving your ball around; what happens to the angles of incidence and refraction?

10.

Do these two methods give you the same solution to the problem? How could you determine this?

11.

Which method do you think is better? Why? Which is more interesting or fun? Which seems ``better"?

12.

Obviously, you can't refer to Geometer's Sketchpad every time you play pool, but how could you use what we did to help you win at pool?