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THE DISSECTED TRIANGLE.

(Various Dissection Puzzles)
A good puzzle is that which the gentleman in the illustration is showing
to his friends. He has simply cut out of paper an equilateral
triangle--that is, a triangle with all its three sides of the same
length. He proposes that it shall be cut into five pieces in such a way
that they will fit together and form either two or three smaller
equilateral triangles, using all the material in each case. Can you
discover how the cuts should be made?
Remember that when you have made your five pieces, you must be able, as
desired, to put them together to form either the single original
triangle or to form two triangles or to form three triangles--all
equilateral.


Answer:

Diagram A is our original triangle. We will say it measures 5 inches (or
5 feet) on each side. If we take off a slice at the bottom of any
equilateral triangle by a cut parallel with the base, the portion that
remains will always be an equilateral triangle; so we first cut off
piece 1 and get a triangle 3 inches on every side. The manner of finding
directions of the other cuts in A is obvious from the diagram.
Now, if we want two triangles, 1 will be one of them, and 2, 3, 4, and 5
will fit together, as in B, to form the other. If we want three
equilateral triangles, 1 will be one, 4 and 5 will form the second, as
in C, and 2 and 3 will form the third, as in D. In B and C the piece 5
is turned over; but there can be no objection to this, as it is not
forbidden, and is in no way opposed to the nature of the puzzle.










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