T

he ra

tio of length

t

o width in this

If y

ou’re not a wiz at

f

rac

tions, y

ou could check

T

he ra

tio of length

to width in this

similarity and congruence

Complex shapes can be similar, too

Similarity isn’t just for triangles! Provided you shrink or grow your shapes

proportionally, they can also be similar. When shapes are proportional,

the ratios between the lengths of their different lines are the same.

T

he ra

tio of length

to width in this

design is 120/56.

120

56

60/28 and 120/56

Similar:

60

28

are the same ratio.

design is 60/28.

this on a ca

lcul

a

t

or.

60/46 and 120/56

Not Similar:

design is 60/46.

60

46

aren’t the same ratio.

Can you use proportionality to tell if shapes

are similar even if you don’t know ANY of their

angles?

you are here 4 73

similarity exposed

Similarity Exposed II

This week’s interview:

Ratios or angles, which is the

real similarity?

Head First: You’re really becoming popular—a lot

of people are saying you’re the time-saving technique

they wish they’d always known.

Similarity: Yes—it’s nice of you to say so! I do like

to think I’m rather, um, efficient is the best word, I

guess.

Head First: That’s certainly true! But there’s one

thing I’m wondering.…

Similarity: Go on.…

Head First: Well, people recognize you by

matching angles—and others use the proportional

thing—and I’m just wondering, which is the real you?

Similarity: I don’t understand. You mean you think

I can only be one or the other?

Head First: Well, surely one is what you’re really

about, and the other is just a convenient alternative

way of presenting yourself. I want to get to the heart

of the real similarity—who are you when you’re just

relaxing at home?

Similarity: Well, to be honest, I really am always

both! I know it sounds silly, but I’ve never thought of

my different aspects as being separate. With triangles,

and a lot of other shapes, too, if the angles are

matching, then the sides are also proportional. I can’t

really pick and choose one or the other!

Head First: And what about if you’ve got

proportionality; if ratios between the lengths of a

triangle are the same, but you don’t have matching

angles? Do you feel something is missing?

Similarity: But that could never happen with a

triangle! That’s just how it is. Anytime triangles have

the same ratios, they have the same angles. You’ve

made me anxious now…but honestly, it’s just not

possible. Proportionality and angles—with triangles

it’s always about both, equally together!

Head First: Together? I didn’t know you were

mixing it up like that. Interesting.… Now, you said,

“a lot of other shapes, too”—that suggests that it’s

not always the case that angles and proportionality

go together?

Similarity: Ah, well, there are some shapes that are

different. Take rectangles for example. All rectangles

have the same angles—90, 90, 90, and 90 degrees.

But they aren’t all proportional—you can have long

skinny ones and short fat ones.

Head First: So you don’t work with rectangles at

all?

Similarity: Oh, I do. But only proportional ones.

Like if you had a rectangle with sides 3 and 6, and

one with 4 and 8—you’d know they were similar.

And squares! I love squares. All of them are similar.

Every single one. Beautiful. Just beautiful.

Head First: Right. Beautiful squares, eh? Thanks

for the interview.

You can spot similarity using

angles or the ratios between

lengths or sides, or both.

74 Chapter 2

similarity and congruence

Based on the old diagram and the angles you’d figured out earlier,

mark up a fresh design to fit Liz’s phone. It needs to be half the

size of the original.

86º

120

56

6060

37.5

30

40

48

40

6

60

41º

53º

86º

139º

53º

41º

127º

94º

139º

41º

41º

86º

53º

127º139º

you are here 4 75

finishing up your new diagram

Based on the old diagram and your angle workings, mark up a

fresh design to fit Liz’s phone. It needs to be half the size of the

original.

All the angles ar

e the same, but the

lengths need to be divided by t

wo.

86º

120

56

6060

37.5

30

40

48

40

6

60

41º

53º

86º

139º

53º

41º

127º

94º

139º

41º

41º

86º

53º

127º139º

20

15

3030

60

28

30

18.75

76 Chapter 2

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