[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
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:
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 <firstname.lastname@example.org>
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 <email@example.com>
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?