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The Geometry Forum Newsletter

Summer 1994, page 3

POM -- Project of the Month

Problem of the Week for April 4-8

Solutions will be accepted from teams of no more than four students. The team submitting the most complete solution will receive Geometry Forum t-shirts. Anything may be used to reach a conclusion, but solutions must be submitted electronically. Computer-constructed documents may be sent -- the answer does not have to be just in text form.

You don't have to prove the answer to submit your solution; just provide a convincing argument for your claim. This may include an explanation with accompanying pictures, or a well-documented sketch in Sketchpad or some other drawing program.

• January 1994: Take any quadrilateral. Construct perpendicular bisectors of the sides to get a new quadrilateral. Repeat on the new quadrilateral to get another quadrilateral. What is true of this new quadrilateral? How can you show that this is true?

• February 1994: What figure is formed when the consecutive midpoints of the sides of a quadrilateral are joined? What if the original quadrilateral were a rectangle? A kite? An isosceles trapezoid? A square? A rhombus? Other shapes? Explain why your answer is true.

• March 1994: For which N's is it possible to make a perfect square using all the pieces from N sets of Tangrams? Example: We know you can make a square with one set of tangrams. If you take all the pieces from two sets, can you make a square? How about with three sets? Four? More? Explain why some of the numbers work and some don't.

  Tangrams are seven polygons that fit together to form many different shapes, one of which is a square. If you don't have access to tangrams, there is available on our ftp site a postscript document that is a picture of the seven pieces. You can print out as many copies of this as you would like to play around with; it's called tangram.ps and can be found in the /project.of.the.month directory.

• March 21-25: One of the most familiar proofs of the Pythagorean Theorem shows a right triangle with squares constructed on each of the edges. The sum of the areas of the squares constructed on the edges equals the area of the square constructed on the hypotenuse. What's so special about squares? What if we used equilateral triangles instead? Or maybe hexagons? Would these figures give the same result? Why do you suppose squares are usually used?

• April 18-22: Given a quadrilateral that has one pair of opposite sides congruent, the other pair not congruent, and a pair of opposite angles that are supplementary, what is this figure?

• April 25-29: Two congruent 6cm x 6cm squares overlap. A vertex of one square is at the center of the other square. What is the largest possible value for the area where they overlap? (The one square is movable, as long as the vertex remains in the center.)

• May 2-6: a) A 6x8 inch paper rectangle is folded so that opposite vertices touch. Find the length of the fold. b) An 8x12 inch paper rectangle is folded so that a vertex touches the midpoint of the longer nonÐadjacent side. Find the length of the fold.

• The Geometer's Sketchpad (for Windows & Mac) and Cabri Geometre (Mac only) can be downloaded as demos that will do everything but save and print. Look in mathforum.org/software/demos.

• A demo version of TesselMania is available from the forum archives as /software/demos/tesselmania.sea.hqx. It's essentially for all Macs (color is recommended). Build the tile and watch construction, complete with transformations.

• For PCs, you can ftp a tessellation demo from mathforum.org as /software/demos/tessl.zip. To extract it, use PKUNZIP.

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