A Math Forum Project

Geometry Forum - Problem of the Week Solutions - Constructing Diverse Quadrilaterals, March 27-31, 1995 Annie says: Interesting responses this week. More than a few people didn't come up with all rhombii - they threw in squares, mostly. While their answers weren't really wrong, they weren't as right as they could be. They were putting extra constraints on the figures they were constructing. I love this problem :-) It stemmed from a Sketchpad workshop I helped out at Haverford High School. Participants were supposed to construct a rhombus as many ways as possible. I thought you could come up with different ways, give each way to a group in a class, and then have the groups report back to the class about what they constructed. It so happens that Ron Hocking did just that, and here's the exchange that occurred after I asked him and his students for some explanation of their answers: I divided our Geometry classes into 4 groups per class and gave each group the given criteria for the quadrilateral that was to be constructed. Each group said that the resulting quadrilateral had to be a rhombus as each had properties unique to the rhombus. In class, each group worked only on the one piece of given information. Then I wrote back: > That's awesome! (That's exactly how I originally developed the idea - to > be done by different groups in the same class.) How did it work out? Were > they surprised? Impressed? Puzzled? Did any interesting discussion > result? (Yes, I'm trying to milk you for information!) > > -Annie Ron responds: Yes, I feel that the discussion went quite well. Each group was surprised that it arrived at the same quadrilateral. I feel that this exercise was very good for several reasons. First, it was a good review of the special properties of a rhombus; second, the exercise did not take much class time; and third, it was a very "workable" problem that most of the students could solve--and thus gain a good feeling of success. Some "puzzles" are so difficult that only the very best of students will even make an attempt. Pretty cool, huh? Anyway, Ron has summed up everything else I wanted to say. Daniel Gabriel Daniel Gabriel Problem of the Week, March 27-31 Akiba Hebrew Academy, Grade 10 All four of the students are constructing the same thing, a rhombus. Natalie is making a rhombus because her quadrilateral has 4 congruent sides and the definition of a rhombus is that it is a quadrilateral with 4 congruent sides. Robert is also making a rhombus because in his quadrilateral, the diagonal bisect each other, making it a parallelogram. Since the diagonals are also perpendicular it is a rhombus because there is a theorem which says that a rhombus is parallelogram with perpendicular diagonals. Dennis is constructing a rhombus because a theorem states that the diagonals of a rhombus bisect the angles. Finally, Jerome is making a rhombus because he is making a parallelogram with consecutive sides congruent. If it is a parallelogram, then opposites sides are congruent, and so all sides are equal making it a rhombus. Therefore, all four students are making rhombuses. Students at Fremont High School The figure that each of the students would construct would be a rhombus. Jeff Beckman, grade 10, Fremont High School Kristi Bennett, grade 10, Fremont High School Roby Bates, grade 10, Fremont High School Jena Bruner, grade 10, Fremont High School These students are the student representatives of Mr. Hocking's Geometry classes at Fremont High School in Fremont, NE. David Gluckwan David Gluckman, grade 9, Fairfield HS From the information given, they are constructing a rhombus because, in a rhombus, all sides are congruent which also means that consecutive sides are congruent. Also, in a rhombus, diagonals bisect the angles and are perpendicular bisectors of one another. Brad Warner Brad Warner, grade 9, Fairfield HS The figure that you are trying to construct is a rhombus. First you draw a segment of any length. Then, using a compass, construct a congruent segment so that both segments have a common endpoint. Next, connect the free endpoints of the two segments. You should have an isosceles triangle. Construct a line perpendicular to the base of the isosceles triangle from the point where the congruent segments meet. It's also a perpendicular bisector. Make sure to extend this segment far past the side it bisects. Next construct two segments congruent to the first two and that have a common endpoint in the perpendicular bisector and each have one endpoint at the vertex of the base angles of the isosceles triangle. You have just constructed a rhombus. Jessica Muniz Jessica Muniz, grade 9, Fairfield HS A polygon is a figure that is formed by three or more coplanar segments called sides. Its sides have a common endpoint that are noncollinear, and each side intersects exactly two other sides, but only at their endpoints called vertices. A quadrilateral is a four-sided polygon, therefore a rhombus is a quadrilateral. The definition of rhombus is all four-sides congruent. A rhombus is a parallelogram because a quadrilateral is a parallelogram if both pairs of opposite sides are parallel. It can be proven that in a parallelogram the diagonals bisect each other. It can be proven that the diagonals of a rhombus are perpendicular and that each diagonal of a rhombus bisects a pair of opposite angles. Therefore, because each of the four figures in the problem describe a rhombus the figure must be a rhombus. Kim Biedermann Kim Biedermann, Mount Saint Joseph Academy, Grade 10 POW March 27-31 Construct a quad with 4 congruent sides: A rhombus was constructed using 2 circles because all radii are of congruent circles are congruent. Construct a quad where diagonals are perpendicular bisectors of each other: The diagonals of a rhombus are perpendicular bisectors. Construct a quad whose diagonals bisect the angles: a rhombus Construct a 4-sided parallelogram with congruent consecutive sides: a rhombus made from 2 congruent circles because the radii (sides of the rhombus) are congruent. Katie Devine & Donna Brennan Christina Carnevale and Kristin Burton Christina Carnevale and Kristin Burton Mount St. Joseph Academy, grade 9 Students in a geometry class were all to construct a quadrilateral. Natalie was to construct a quadrilateral with four congruent sides, Robert was to construct a quadrilateral where the diagonals are perpendicular bisectors of each other , Dennis was to construct a quadrilateral whose diagonals bisected the angles, and Jerome was to construct a four sided parallelogram with congruent sides. Conclusion: The physical features of the rhombus will satisfy all of the requirements for each student. No matter how the student draws the figure according to their directions, it will be a rhombus. Depending upon how each student shapes their rhombus, a square may result, also having the specific properties required for each student. We chose to use the term rhombus because "rhombus" is a more general term than "square". Although the figure drawn by each student must be a rhombus, it may not be a square. A square is always a rhombus, but a rhombus may not always be a square. RHOMBUS' PROPERTIES SQUARES' PROPERTIES -- parallel opposite sides -- all the properties of a rhombus, plus -- four congruent sides -- all four angles are 90 degrees. -- diagonals are perpendicular -- diagonals bisect each other -- diagonals bisect the angles Marianne Ganster, Kathy Diamond, Alicia Kempf, Siobhan O'Brien The quadrilateral that we constructed with four congruent sides, the diagonals are perpendicular bisectors of each other, the diagonals bisect the angles and the consecutive sides are congruent is a rhombus. The group referred to in the POW is 10,000 Maniacs. Susanna Puntel, Megan Nugent All of the groups in the problem have been asked to construct the same quadrilateral - a rhombus. The rhombus that we constructed fits all of the elements. All four sides are congruent and therefore have congruent consecutive sides. The diagonals are perpendicular bisectors of each other and the diagonals bisect the angles. The rhombus which we constructed is neither over or under constructed. While the measurements can change, the properties remain constant. Previous page || Next problem || Previous problem || Table of Contents || Forum Home Page