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 and then to investigate an important and useful property of parabolas.
1.
Start by creating a horizontal line 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.
3. Hide points A and B.
4. Create a free point P on the directrix. Slide point P a little to the right of the center of the line.
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 do not need to include this
note on your document. The sketch below is just for explanation purposes:
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. Start by creating 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.
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 we called t (and above segment FP). Note that
E is equidistant from point F and from point P.
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).
How can we argue that point Y is INDEED on the parabola that we wish to create?
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 just 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!
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. And time to 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.) We also
used our understanding of how we measure distance from a point to a line. 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.
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 a tangent line as well. Click on the arrow tool and the
open space to make sure nothing is highlighted. The, 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 consider the light ray that hits point Y, and let’s make the light ray come in perpendicular
to the directrix. To do this, 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 it is going to go after it hits the mirror.
15. Well, remember “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 dish is angled at point Y where
the light ray is hitting the parabola. 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 perpendicular line that is serving as our light ray and choose Reflect from the
Transform menu. You should see your reflected light ray.
What is it that is REALLY INTERESTING about where this light ray is going to go once it bounces
off of the reflective parabola?
16. Click on the arrow tool and then open space. Next, drag point P again back and forth along the
directrix. Observe all of the light rays that come from the sun.
Do all “light rays” bounce to the same place when they reflect off of the parabolic mirror?
17. Now, think about the significance of this in terms of solar power? If we can place an object at
the focal point that can capture the light rays and absorb the heat, we are continuously
capturing the energy from the sun. This heat energy can be transformed to other sources of
energy in order to do work. (No need for the burning of coal and oil to generate heat and
harness energy. Pretty simple idea! European governments have figured this out and
promoted solar energy production. Why the US has failed to promote the harnessing of energy
from the sun above any other means of generating energy is hard to understand. Instead, let’s
just keep spending tremendous amounts of money to dig very deeply into the Earth in areas
that otherwise would be beautiful wildlife refuges and pull out dirty oilish or coal deposits that
need to be cleansed and then transported elsewhere taking huge amounts of energy and
producing much CO2 in the process contributing to global climate change and then burned
creating still more CO2 and contributing even more to climate change …. Those oil company
lobbyists with deep pockets are the ones that should be ….. Oh ….. uh ….. – sorry for the
digression. (And, to be fair, the oil lobby counters with the argument that we need to gain
independence from foreign sources – which indeed needs to be part of the equation. But we
could debate this until the cows come home so instead let’s go back to the math.)
18. Believe it or not, there is at least one hand held item in your house that is powered by a battery
that uses the reflective property of a parabola to perform a task. Whether you can figure that
application out or think of (or research) another, 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.
GEOMETRY: PARABOLA LAB ANSWER SHEET
10. How can we argue that point Y is INDEED on the parabola that we wish to create?
15. What is it that is REALLY INTERESTING about where this light ray is going to go once it bounces off of
the reflective parabola?
16. Do all “light rays” bounce to the same place when they reflect off of the parabolic mirror?
18. Your final task is to describe one other real world application of the use 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.