ULTIMATE MATH
DRAW THE LINE
HOW YOU SLICE IT
Let’s start things off small and simple, with a hexagon. A
diagonal is a line that joins two nonadjacent corners of a
polygon. The nine diagonals of a hexagon appear in red.
To dissect a figure is to cut it into pieces, like the
parts of a jigsaw puzzle. The hexagon below has been
dissected into four
triangles by three
diagonals. Note
that the diagonals
do not intersect.
1. How many
different ways are
there to dissect
a hexagon into
triangles using only
nonintersecting
diagonals?
(Dissections that differ by rotation or
60°
reflection are considered different.
For instance, the hexagon above
rotated by 60° or 120° would
constitute a different dissection. If
rotated by 180°, however, it yields
the same configuration as the one shown, so that would
not be considered a different dissection.)
2. Now go further: How many triangles can you cut a
pentagon into using nonintersecting diagonals? How
many for a heptagon?
1. How many different diagonals can you draw in the
pentagon? In the heptagon?
2. To get a handle on abstract problems,
mathematicians examine specific examples and look
for general patterns. Without drawing anything, can
you predict how many diagonals you can draw in a
nonagon? A decagon? Hint: Look at the number of
diagonals in a square, a pentagon, a hexagon and so
on, and try to find a pattern.
3. Without drawing anything, can you predict how
many triangles you can cut an octagon into, using only
nonintersecting diagonals? A nonagon? Hint: Again,
look at the pattern in the number of triangles for a
square, a pentagon, a hexagon and so on.
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