90

Ha

nd dr

aw the

sun and r

ays

what does “the same” mean?

The design tells us that some triangles are repeated

120

The designer made a note saying that the two mountain

T

hese notes ar

e a big hint

about the missing informa

tion.

6060

37.5

30

40

48

40

6

60

41º 53º

This corner a

lso

Mou

ntain

tip tria

ngle

is the same

shape

tria

ngle as

the whole

mou

ntain

T

h

e

t

w

o

m

o

u

n

t

a

i

n

t

ri

a

n

g

le

s

a

r

e

t

h

e

s

a

m

e

s

h

a

p

e

,

bu

t

r

e

s

c

ale

d/flipped

56

triangles are the same but rescaled, or different sizes.

There’s also a note saying that the mountain tip is the same

triangle as the whole mountain.

But what exactly does he mean by “the same triangle”?

same?

Jim: Yeah, I mean, the designer is talking nonsense. One of

them is bigger than the other. How on earth can they be the

Frank: Well, maybe he was talking about angles. Can you have

two triangles with the same angles but with different lengths?

Same triangle?

I don’t get what

this means.

Frank

Jim

54 Chapter 2

Jim

similarity and congruence

GEOMETRY Construction

Can you have two triangles with the same angles but with different lengths?

1) Cut or tear three narrow strips of paper, making them slightly different lengths.

2) Make them into a triangle and draw around it on some scrap paper.

3) Now fold each of your strips of paper in half, join them up to make a triangle again, and draw around it.

Compare your drawings to investigate what happens to the angles of a triangle when you make it bigger or

smaller—you just need to make sure that you do the same to each side of your triangle.

Does your investigation help you to fill in any of the mystery angles on the design?

41º 53º

?

??

?

?

?

?

you are here 4 55

geometry construction solution

GEOMETRY Construction SOLUTION

Can you have two triangles with the same angles but with different lengths?

1) Cut or tear three narrow strips of paper, making them slightly different lengths.

2) Make them into a triangle and draw around it on some scrap paper.

3) Now fold each of your strips of paper in half, join them up to make a triangle again, and draw around it.

Compare your drawings to investigate what happens to the angles of a triangle when you make it bigger or

smaller—you just need to make sure that you do the same to each side of your triangle.

Does your investigation help you to fill in any of the mystery angles on the design?

?

53°

41°

C

hanging the size of the

triangle doesn’t change

the angles.... So each

triangle mus

t hav

e the

same angles as this one.

W

e don’t know

this angle.

We know these

t

w

o angles.

Making a triangle bigger or smaller doesn’t change the angles of the corners—

providing you change the length of all the sides by the same ratio.

The mountains are basically made out of 4 of

the same triangle in different sizes (one tucks in

the back but the other corners are the same).

1

2

3

4

56 Chapter 2

GEOMETRY Construction SOLUTION

similarity and congruence

you are here 4 57

Hold on—didn’t you spend Chapter 1

going on about how we couldn’t trust our

eyes to tell us if things looked right?

Now you’re saying these triangles are

“similar”? Sounds flaky to me—what’s the

deal?

180

o

in a tr

iangle

(180

o

-(41

o

+ 53

o

) = 86

o

F

ir

s

t find the missing

angle in the tr

iangle…

…and then copy those angles t

o

a

ll f

our of the tr

iangles, as w

e

know they’r

e a

ll the same.

T

he f

our triangles have the same

angles because they’re simil

ar.

86°

53°

41°

41º 53º

86º

86º

41º

41º

41º

53º

53º

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