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January Project of the Month Winners!
Posted:
Feb 22, 1994 3:56 PM
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[My apologies for being so long in posting this. We experienced some
transmission problems with a couple teachers - I received blank or otherwise
unreadable submissions from them at the end of January, and have spent a few
weeks working things out so that their students' solutions could be included.
Finally, we're all set, and hopefully these problems won't turn up in future
months. My thanks to the teachers who stuck with it - the first time is the
worst. -Annie ]
**********************
Some comments on the judging process: While we were looking for the right
answer, we were really looking at the process of solution, and how much
investigation the group went through to try to provide a convincing argument
for their claims. These problems are hopefully designed so that students can
play around with them in trying to figure out "why".
One thing missing from a lot of the entries was _how_ they gathered their
information. It's helpful to people reading a solution to know if pictures
were draw, how they were drawn, if something like Sketchpad was used (and if
so, if the drawings were included!), and how many different drawings were
used. Just as in science experiments, process is very important in evaluating
a solution.
Below we have included three of the submissions. The first is the winning
entry, and the two others gained honorable mention. The reasons for our
choices are included below each entry. It is my hope that these comments will
give teams a bit of guidance in preparing well-reasoned submissions for both
our Project of the Month and their own work in the future.
The winners will each receive a Geometry Forum tshirt and a certificate.
Honorable mention entries will receive a certificate.
All other correct entries will be included in a message to follow this one.
These will also have some comments attached to them.
All these files can also be found on our archives, in the directory
/project.of.the.month/solutions/january.94
I have also included in that directory a Sketchpad sketch that I did of the
problem. It's not that polished right now, but it's the sort of thing _I_
would like to see as part of a solution to this problem (not to mention the
fact that it helped me to understand the problem/solution better).
**********************
Now, for the winners! January's top entry was submitted by:
From: crags@delphi.com
Jennifer Burrows, Bonnie French, Caroline Samponaro, and Pam Schoenberg
of Greenwich Academy, 200 North Maple Ave.,
Greenwich, CT 06830
Their solution follows:
In this essay our goal was to prove that any random quadrilateral's
perpendicular bisectors would produce another figure whose perpendicular
bisectors would produce yet another quadrilateral which is similar to the
original quadrilateral. To prove this statement correct, it was necessary for
us to apply many of the concepts we have learned throughout the year.
The first step we took to prove the quadrilaterals similar was to measure all
of the angles in the first and third quadrilaterals. We found that each angle
of the original figure had a corresponding angle on the last figure. Although
this alone does not prove the two to be similar, it is very important
information.
This not only satisfies part of our hypothesis, it shows us which sides
correspond with each other. By using the above information, we set up the
corresponding sides in proportions and calculated the ratio of their distances
by dividing the distance of a side of the third figure by the corresponding
side of the original figure. We repeated this process for each pair of sides
and found that the ratio, 0.18, was maintained for all four pairs of sides.
This information and the equal angles proved the two figures similar.
However, we also found a different and more original approach to the same
situation. The term perpendicular bisector is an important, yet sometimes
overlooked one. Each perpendicular bisector of the different sides of the
original quadrilateral have two constants. The first is that a perpendicular
bisector always cuts the segment exactly in the middle, so both of the two new
quadrilaterals will have the same ratio of lengths of sides. The second is
that the lines are perpendicular. Two lines perpendicular to a third line
are parallel. The perpendicular bisector of the original figure has a slope
that is the negative reciprocal of that line. The line perpendicular to the
perpendicular bisector of the first quadrilateral also has the negative
reciprocal slope, but this slope is the same as that of the original figure,
therefore the lines are parallel. Even though the lines are parallel, the
each figure is rotated 90. Therefore the third figure will be rotated 180
from the original figure.
In conclusion, we have proved the first and third quadrilaterals to be similar
using two different methods. In the first proof, we showed that this was true
with the figure we were using, and in the second, we proved that this will be
true with any random quadrilateral. By gathering this information we have
solved this complex problem.
COMMENTS: While a number of groups came up with the similarity, this group
was the only one who really tried to explain _why_ the figures were similar.
It would be a good idea to provide information about certain kinds of special
quadrilaterals in future problem (as the groups below did, a bit).
**********************
Honorable mention #1:
From:Julie Jadlowiec <jadlowie@one.sasd.pa.us>
Hello! We are Julie Jadlowiec, Eric Carlson, Alyssa McGrath, and Jim
Sadowski. We are in a tenth grade honors geometry class at Shaler
Area High School in Pittsburgh, Pennsylvania. These are our results
for the following problem of the month.
Take any quadrilateral. Construct the perpendicular bisectors of each
side to get a new quadrilateral. repeat on the new quadrilateral to
get another quadrilateral. What is true of the new quadrilateral? How
can you show that this is true?
[They tried to include a tiff drawing with their answer, but we never
had a successful transmission.]
In the figure shown, the quadrilateral is convex. We have drawn all
the perpendicular bisectors of the original figure and found a
second, which is also a quadrilateral. Then we repeated the process
to form the third and final quadrilateral. We discovered that this
final quadrilateral was not only the smallest but was similar to the
first and second quadrilaterals. We have not yet begun to study
similarity in our class but read ahead on the subject to find what
else could be concluded. Similarity, we found, was not the same as
congruence because the sizes of the figures were different. We then
found that the concept of similarity was related to proportionate.
Therefore the first, second, and third figures were not only similar
to each other but also were proportionately related.
The second and third figures we found were not oriented in the plane
the same way.The second figure is rotated 90 degrees to the left. The
third figure is rotated another 90 degrees to the left from the
second figure for a total of 180 degrees from the original figure. We
then measured the corresponding sides of the first and the final
figures using our draw program, Geodraw.We found that the final
figure was 1/3 the size of the original. This would make the
figures related proportionately. Being related proportionately is a
necessity for two figures to be similar. We then measured the
corresponding angles, also using Geodraw, and found them to be equal.
Angles being equal is another method of proving similarity between
figures.
Other results from different types of quadrilaterals:
-If a rectangle is used for the quadrilateral, and the
perpendicular bisectors are constructed, the result is one point in
the exact center of the figure.
-If the quadrilateral chosen is non-convex, the four
perpendicular bisectors drawn do not form a similar quadrilateral.
When the perpendicular bisectors are drawn again, however, the figure
formed is similar to the original. Information about non-convex
figures is incomplete to us at this time.
Our conclusion is that for a convex quadrilateral, the final figure
constructed is similar to the original figure drawn.
COMMENTS: This team talked about the relationships between the original and
resulting figures, including the fact that they were related by 180 degree
rotation and were similar, and then tried out a couple specific
quadrilaterals, but never tried to explain _why_ they are similar. This is
the only shortcoming of their submission compared to the winners. It's often
one thing to state an answer, but another to explain it, or to convince
someone else that it's true.
**********************
From: kate rusbasan <rusbasan@one.sasd.k12.pa.us>
For each of the following, it is assumed that the quadrilateral is in a single
plane:
1. If the quadrilateral is a rectangle (square) then the perpendicular
bisectors form an intersection at one point.
2. Using a parallelogram, the last quadrilateral formed appears to be a
scale model of the original parallelogram.
3. Using a kite, the shape of the new kite is reversed in orientation by 180
degrees.
4. Using a non-isosceles trapezoid, a shape is produced which is the same as
the original and it appears to be a 180 degree rotation counter-clock wise or
clockwise, but the shape is smaller.
Using an isosceles trapezoid, the perpendicular bisectors of the sides will
all intersect at one point.
5. Using a non-convex quadrilateral, the quadrilateral produced is also a
non-convex quadrilateral. It also slightly resembles the first quadrilateral.
6. It appears that using a large quadrilateral as the original shape, the
shape formed by the perpendicular bisectors is smaller than the original.
In all examples above, the cases have been proven by trial and error through
drawing.
By: Mel Monteleone, Tammy Manski, Erin Fisher, Kate Rusbasan
Shaler Area High School
Grade 10
COMMENTS: Often in doing a problem like this, it's an excellent idea to start
off by looking at different kinds of figures - quadrilaterals in this case -
and seeing what can be generalized from the results. However, this team
failed to generalize, and in fact never mentioned that similarity was present.
In 5, they say that the shape "slightly resembles the original". How much is
slightly? This is a good time to see how much it really resembles the
original. And in 6, when they say that the shape is smaller than the
original, they should see if they can find a counter example to this. Can
they construct a shape where it's _not_ smaller? If so, then what is the
relationship between the shapes if it's not size?
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