Cutting a Triangle into Two Congruent TrianglesDate: 10/06/98 at 14:59:33 From: David Auerbach Subject: Cutting a triangle into two congruent triangles I need to know how to cut a triangle into two congruent equilateral triangles with the minimum number of cuts. We have tried trial and error and we got nothing. I would also like to know how to cut a square into two congruent squares with the minimum cuts, because, even though we solved it, our math teacher says there might be a better way. Thanks for the help, -David Date: 10/06/98 at 16:33:17 From: Doctor Rob Subject: Re: Cutting a triangle into two congruent triangles For the squares, two diagonal cuts should suffice, from corner to opposite corner. The triangle is a nice problem. Start with the triangle like this: ______________________________________________________________________ Cut 1: Vertical, through C, intersecting AB at D: ______________________________________________________________________ Flip the left triangle over, and move it so that AC coincides with CB. That makes a rectangle: ______________________________________________________________________ Now cut with a line through D making a 45-degree angle with BD, and meeting BP at E: ______________________________________________________________________ Now move triangle BDE so that BD coincides with PC, to form a parallelogram: ______________________________________________________________________ Now cut with a line through E making the 120-degree angle DEG, and meeting CF at point G. ______________________________________________________________________ Translate triangle EFG so that EF coincides with CD: ______________________________________________________________________ Finally, cut along line EH. Then you have two equilateral triangles DEH and GEH, each half the size of the original. ______________________________________________________________________ This took four cuts. I have no proof that this is minimum, but I am fully convinced that it is. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/ |
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