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Splitting shapes (Posted on 2006-02-16) Difficulty: 3 of 5
1. Divide a square into three similar regions, ALL THREE of which are congruent.
2. Divide a square into three similar regions, EXACTLY TWO of which are congruent.
3. Divide a square into three similar regions, NO TWO of which are congruent.
4. Divide an equilateral triangle into three similar regions, ALL THREE of which are congruent.
5. Divide an equilateral triangle into three similar regions, EXACTLY TWO of which are congruent.
6. Divide an equilateral triangle into three similar regions, NO TWO of which are congruent.

If necessary, you may use one or more inversely similary regions where all corresponding angles are equal and described in the opposite rotational sense, i.e. "reflected".

  Submitted by Bob Smith    
Rating: 4.0000 (3 votes)
Solution: (Hide)
Square
Split into rectangles
1) 3x(1/3 x 1)
2) (2/3 x 1), 2x(1/3 x 1/2)
3) Approximate values: (.57x1),(.75x.43),(.43x.25)

Triangle:
4) Cut shape along any 3 lines from center separated by 120 degrees.
5) This is the tricky one to explain without a picture. Draw a parallelogram in one corner that's about 1/3 of the height and 1/2 the base of the triangle. This parallelogram can be split into two congruent pieces that are similar to the remaining piece of the triangle.
     /\ 
    /  \
   /___ \
  /   /  \
 /___/____\ 

6) Split into 3x(30,60,90) triangles. Cut from a vertex to center of opposite side then from that point perpendicular to one of the other sides.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
5, illustratedDej Mar2006-03-17 04:18:50
re(3): All but 5Jer2006-03-16 13:12:10
re(2): All but 5Dej Mar2006-03-06 15:39:40
Hints/Tipsre: All but 5Bob Smith2006-02-21 11:01:56
SolutionAll but 5Tristan2006-02-17 23:28:01
4 region solutionsJer2006-02-17 12:49:34
Some Thoughtsgoing furtherJer2006-02-16 13:33:42
SolutionPictures of solutionsJer2006-02-16 13:19:09
Hints/TipsParts 1 & 4: Three congruent partsFederico Kereki2006-02-16 12:03:32
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