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Not this sort
of bounding
Her
e’s the top
triangle properties
A triangle fits inside a bounding rectangle
A bounding rectangle is a box that fits tightly
around the triangle.
To draw a bounding rectangle:
1
Pick a side of your triangle to share
with the bounding rectangle.
They shar
e
this side.
(opp
osite) v
ertex.
2
Draw a line parallel to this side,
touching the opposite vertex.
3
Draw two perpendicular lines,
one at each of the other
vertices.
How does the shaded area of triangular field A compare with
This box is
the bounding
r
ec
tangle.
the area of the dashed bounding rectangle surrounding it?
Scale: 1cm = 30m
60m
Field A
90m
you are here 4 157
triangle area is half
How does the shaded area of triangular field A compare with
the area of the dashed bounding rectangle surrounding it?
Scale: 1cm = 30m
60m
90m
Field A
1.
There’s a shaded triangle representing the
Field A
109m
60m
part of the rectangle which is the field.
90m
These tw
o
And there’s an unshaded triangle representing the
tr
iangles ar
e
part of the rectangle which isn’t the field.
If the triangles are congruent, then they must
both have the same area, so they share the
Not
field A
90m
60m
109m
congruent.
2.
Both methods giv
e y
ou the
rectangle area equally—half each.
You might have recognized visually that each bit of
As they are both the same,
the rectangle is shared
equally—half each.
=
THE AREA OF THE TRIANGLE IS HALF THE AREA OF THE BOUNDING RECTANGLE.
158 Chapter 4
ar
en’t r
ight tr
iangles.
triangle properties
Q:
What if I’ve got a triangle which doesn’t have a horizonal
side? How do I draw a bounding rectangle?
A:Use a vertical side if you have one because it does make
things easier, but if you don’t have one, then any side will do.
These ar
e
both fine.
Q:
We’ve talked about sides and angles being the same
in congruent triangles, but how do you know that congruent
triangles have the same area, too?
A:With a bit of flipping and rotation you can always place
congruent triangles on top of one another, and they fit perfectly—so
they must have the same area.
Q:
What if I don’t know the size of the bounding rectangle?
A: Good point. We’re going to cover some ways to find that out.
For RIGHT triangles, this
half-the-rectangle thing
looks sensible. But what
Good question—two of our
fields are not right triangles.
So what can we do?
A
B
C
T
hese t
w
o fi
elds
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geometry detective
Rec
tangles ar
e easi
er to draw if y
our
v
ertic
es are a
ll on p
oints on the gr
id.
Her
e’s one t
o
s
tart you of
f.
160 Chapter 4
Geometry Detective
Using a ruler, draw some non-right triangles. Use a horizontal or vertical
grid line for one side (not two) of each triangle, and make the side that’s
on the grid line the longest side.
Draw a bounding rectangle around each triangle. How does the triangle
area relate to the rectangle area?
triangle properties
Draw at least three more
triangles on these pages
Need a 10-minute time-out?
If you’re feeling kind of fuzzy about
how your triangle areas relate to rectangle areas
try having a 10-minute brain-break, then look
again before you flip the page.
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