GEOMETRY LAB: EXPLORING PARABOLAS
In class we investigated the definition of parabola. Namely, the set of all points equidistant from a
fixed point (called the focus) and a line (called the directrix).
Today, we’ll use Sketchpad to create a parabola. Then, we’ll investigate an interesting property of
parabolas that leads to many real-world applications!
1.
Open a new Sketchpad file. Then, create a horizontal line (be sure to choose the line tool, not
the segment tool) near the bottom of the page. Label the control points on the line as A and B.
With the text tool still selected, click on the line itself and label the line with a lowercase d (for
directrix). (Likely, the line will initially be called “j”. Just double click on the “j” and you will be
able to change the label to d.)
2. Next create a point a little above the directrix that will serve as the focus (about an inch or two
above the directrix). The focus should be approximately centered above the directrix as well.
Label the focus as F. Also, Create a free point P on the directrix.
3. Hide points A and B.
4. Slide point P along the directrix to a location a little to the right of the center of the screen.
Then create a segment from the focus to point P. Also, highlight this new segment and
construct a midpoint. Label the midpoint as M. Your image should look something like the
following:
F
M
d
P
Note: Point M is NOT equidistant to the focus and to the directrix. And hence, point M is NOT
on the parabola that we will be constructing. Remember, we measure the distance from point
M to the directrix along a perpendicular to the directrix. Hence, point M is closer to the directrix
then it is to the focus. See the note below just for clarification. You need not add those
comments to your Sketchpad file.
5.
Remember, our goal is to create points that ARE the same distance from the focus and directrix.
We will use point M and segment FP to do this. Next, create a perpendicular to segment FP
through point M. To do this, highlight point M and segment FP and choose Perpendicular Line
from the Construct menu. Note that this is the perpendicular bisector of segment FP.
6. Label the new line as line T. See below.
t
F
M
P
d
7. Now, recall a theorem that we investigated and proved earlier this year:
Any point on the perpendicular bisector of a segment is equidistant to the endpoints of the
segment.
Let’s plot a point E on the perpendicular bisector (line t) and above segment FP. Note that E is
equidistant from point F and from point P. Again, the diagram below is just for clarification. You
need not add those comments to your Sketchpad file.
And note, this also does NOT mean that point E is on the parabola. It is NOT equidistant to the
focus and directrix. (It is equidistant to points F and P as noted above).
Your diagram should look approximately as follows:
t
E
F
M
d
P
Just to be clear, we still haven’t plotted any points that are equidistant to the focus and the
directrix. In other words, we have not yet plotted a point on the parabola that we wish to draw.
8. But now, our diagram is set up to create points on the parabola. Well, almost. We are done
with point M and E now. Hide points M and E.
9. Next, let’s create a perpendicular to the directrix through point P. To do this, highlight point P
and the directrix and then choose Perpendicular Line from the Construct menu. Label the line as
p (for “perpendicular” to directrix).
10. Plot a point at the intersection of the new line p and line t. Label the point as point Y. Note that
point Y is INDEED on the parabola that we wish to create. (The letter Y is for “Yes” we have
finally plotted a point on the parabola). Your diagram should look approximately as follows:
How do we know that point Y is INDEED on the parabola that we wish to create? Your answer
should be written with a complete sentence or two. See the two hints below!
(Hint #1: Point Y is on line t, the perpendicular bisector of segment FP. So Y is equidistant to
what two points?)
(Hint #2: Point Y is also on line p, which is perpendicular to the directrix. Why is this
important?)
<type answer here>
11. Click on the arrow tool and then open space to make sure nothing is highlighted. Next, drag
point P back and forth along the directrix and watch the path of point Y. Do you see that Y is
tracing out a path along a parabola? Wouldn’t it be great if we could “trace” its path to actually
see the parabola? Well, we can!
To do this, highlight ONLY point Y and then choose Trace Intersection from the Display menu.
Next, click on the arrow tool and then open space to deselect everything. Next, drag point P
back and forth along the directrix and watch as Sketchpad traces the path of point Y to form a
parabola! (If something goes wrong, you can erase the traces with the display menu).
12. To get ourselves out of any potential future troubles, we want to turn the trace off now. To do
this, click on the arrow tool and then open space. Next, highlight just point Y. Now, from the
Display menu, you will see a checkmark next to the Trace Intersection. Click on Trace
Intersection again. This will serve to turn off the Trace. (You do NOT want to erase the traces.
We want to keep the trace of our parabola on the screen.)
OK – time for a very short intermission. Take a minute or two and relax while looking at this nice, calm
image.
Let’s reflect briefly on what we have done. In a very creative way, we found a way to construct point Y
to be equidistant to the focus and directrix (by using the intersection point of a perpendicular bisector
and a perpendicular line to the directrix.) In the end, we have a traced sketch of a set of points
equidistant from a focus and a directrix. In other words, we have a great sketch of a parabola. Take one
last glimpse of the calm image above, and then let’s continue!
Now, for the second half, we will investigate a reflective property of a parabola. This reflective property
leads to many important and valuable applications of parabolas in the real world. Now, onward!
13. We are done with the perpendicular line p and also with segment FP. So, take the time to hide
both of these straight objects. Your sketch should looks as follows:
14. There is a reason we labeled one of the lines as “t”. It is not only circles that have tangent lines.
A parabola (or any other curve) can have tangent lines as well. Click on the arrow tool and the
open space to make sure nothing is highlighted. Then, drag point P back and forth along the
directrix and watch line t. Just like the tangent to a circle, line t only intersects the curve at one
point. Point Y is not only a point on the parabola, but also at all times a point of tangency.
Now, let’s imagine that our traced parabola is a reflective material serving as a solar panel and
pointing upward toward the sky. Now, let’s bring a light ray from the sun into our solar dish.
Let’s make the light ray come in perpendicular to the directris. And, let’s consider a light ray
that hits point Y. To create such a light ray, highlight the directrix and point Y and then choose
Perpendicular Line from the construct menu.
Note that the light ray is not going to pass through point Y as our line does in the sketch.
Instead, the parabola is going to be made of a reflective material (serving as a mirror). When
the light ray comes down from the sun and hits point Y, it is going to reflect off the mirror. We
just need to figure out where the light from the ray is going to bounce after it hits the mirror.
15. Well, remember the saying “angle in equals angle out”? The “angle in” is the angle between
the light ray and the tangent line. Note that the tangent line tells us how the parabola is angled
(or bent) at point Y where the light ray is hitting the parabola. Hence, the tangent line at point Y
can serve as the mirror for a light ray hitting at point Y. So, let’s reflect the light ray over the
mirror (in other words over line t) to see where it goes. To do this, double click on line t to mark
it as the mirror. Next, select the line that represents the light ray and choose Reflect from the
Transform menu. You should see your reflected light ray. See clarification below:
Click in the open space so that nothing is highlighted. Now, drag point P back and forth. You
need to imagine that each line coming down from above is a light ray. The light ray strikes the
inside of the parabola and bounces in another direction. But, every single light ray coming from
above has something in common. What is noteworthy about where each light ray that comes
from above is going to go once it strikes the parabola?
<type answer here>
16. Save your GSP file to your SSP Folder in your H drive. Name the file as “Geometry Parabola
Lab”.
17. Now, think about the significance of this reflective property of a parabola in terms of solar
power. Imagine that we have parabola shaped “dish” with reflective material that is pointing
upward to the sky. Now, let’s place a “receiver” at the focal point that can capture the light rays
and absorb the heat. Any light ray that strikes the inside of our dish will be re-routed to the
receiver. The receivers immediately get very hot. This heat energy (coming continuously from
the sun) can be transformed to other sources of energy in order to do work. (No need for the
burning of coal and oil to generate electricity. The heat from the sun, once captured, can be
used to produce electricity.) Pretty simple idea! European governments have figured this out
and promoted solar energy production. Europe is filled with fields of solar panels like the field
shown below. The U.S. lags far behind in the promotion and use of solar power. But progress is
being made, especially in the western states.)
18. Your final task is to describe one other real world application that uses the reflective property of
a parabola to perform some useful task. Make sure to explain the role of the focus in your
application and the role of the parabolic shape as well. Also be sure to use complete sentences.
<type answer here>