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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