Head First 2D Geometry

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

There are two different ways to think about this question....

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

same answer: half.

rectangle area equally—half each.

You might have recognized visually that each bit of

As they are both the same,

shaded cm square grid has matching unshaded bit.

the rectangle is shared

Counting up the shaded and unshaded squares:

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

about non-right triangles?

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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