Greenwich Academy, Greenwich, CT

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 that 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).

Tenth grade honors geometry class, Shaler Area High School, Pittsburgh, Pennsylvania

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.

Grade 10, Shaler Area High School, Pittsburgh, Pennsylvania

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.

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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2 July 1995