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Head First 2D Geometry by Dawn Griffiths, Stray

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triangle properties
Let’s put the drink stall at the centroid
Outside the venue would be a pretty weird place to put your drinks stall, so it’s
important to pick a center which is always interior.
The centroid also never gets pulled particularly close to one side or vertex,
whereas the incenter can end up very far from one vertex if the triangle has
sides of very different lengths.
An interesting thing about the centroid is that it’s where the triangle
balances—this means that each median splits the triangle area into two equal
halves.
Stage a
t this end.
Scale : 1cm = 10 meters
Using accurate construction, find the position of the centroid of
your field to the nearest meter.
Do you notice anything interesting about the distance from the
centroid to the stage?
and ruler
you are here 4 187
find the centroid
Using accurate construction, find the position of the centroid of
your field to the nearest meter.
Do you notice anything interesting about the distance from the
centroid to the stage?
and ruler solution
50m
75m
Stage
this
end
Scale : 1cm = 10 meters
25m
The centroid is exactly 2/3 of the
way from the stage to the back of
the venue.
To find the c
entroid, find the
midp
oint of each side and draw
a line t
o the opp
osite v
ertex.
Q:
I drew an equilateral triangle
and my centers were all in the same
place—did I do it wrong?
A:That’s perfect! All the four centers
of an equilateral triangle are in the same
place. It’s the only type of triangle that
truly has a “middle.” Centers of an isoceles
triangle all fall on the altitude of the
base—the line you might use to find the
height.
Q:
Do I really need to know four
different centers—won’t one do?
A:There are actually over 3,500! These
are the four you’ll use the most, though.
The centroid is used to balance a triangle
in the physical world, but the other three all
have uses. Each has different potential to
find other things—a line from the incenter
to any vertex of a triangle bisects that
angle at that vertex, so you can use the
incenter to find that angle if you need to.
Q:
Why on earth would a center that
lies outside the triangle be useful?
A:The orthocenter and circumcenter
only lie outside the triangle sometimes.
If your triangle is acute then you might
have need for a center that is exactly the
same distance from all three corners, or
one where a line to any side meets it at a
right angle. It’s totally dependent on what
triangle you’ve got and what you’re trying
to do with it.
188 Chapter 4

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