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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!
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 youd 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
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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