Head First 2D Geometry

diff

er

ent sizes of gadgets.

similarity and congruence

Ratios can be more useful than sizes

A ratio captures the proportions of a shape, and

then by using a different factor, we can create a

similar shape of any size.

T

his is an important step.

T

his bi

t is easy—i

t’s jus

t

multiplica

tion or division.

By using a diff

er

ent f

ac

t

or,

her

e w

e can make the design fi

t

T

his is wha

t y

ou

need t

o

etch

wi

th.

Factor to multiply

EVERY length by

5

12

3

Diagram with

correct ratios

Diagram with

correct lengths

The notes the drummer made on the diagram tell you three

ratios—what are they? Draw sketches if it helps you figure it out.

All the arr

ows ar

e the same.

The squ

are par

t is ha

lf the

length of the arr

ow head par

t.

T

riangle sides ar

e the same.

The darker set of arr

ows ar

e

3/4 size.

you are here 4 93

sharpen solution

The notes the drummer made on the diagram tell you three

ratios—what are they? Draw sketches if it helps you figure it out.

3:4

2:1

2

1

1:1

1

1

1

3

4

All the arr

ows ar

e the same.

The squ

are par

t is ha

lf the

length of the arr

ow head par

t.

T

ria

ngle sides ar

e the same.

The darker set of arr

ows ar

e

3/4 size.

Q:

How come 1:1 is a ratio? Isn’t that a bit pointless?

A:It does sound a bit weird, doesn’t it? 1:1 is a way of

indicating that it’s exactly the same size as the original. If you

didn’t put the 1:1 in there, then someone reading your work or

diagram might wonder whether you’d forgotten to put a ratio in for

that item. 1:1 indicates clearly that it’s the same size.

Q:

The drummer wrote that the dark arrows are 3/4 size,

so why have you written it as 3:4?

A:3/4 and 3:4 are just different ways of indicating the same

relative proportion. 3/4 is fraction notation and 3:4 is ratio notation.

Unless you’ve been specifically told to use one or the other then

they’re mostly swappable.

94 Chapter 2

similarity and congruence

Uh, sorry, dude, but you’ve messed

up. You’re saying that the arrow heads

are triangles with size 3, and size 4, and

size 2…well, they can’t be ALL of them!

Jim

Joe

Frank: I kinda followed what you did though…with the

ratios, and it all makes sense.

Jim: I don’t think so, it has to be wrong. There’s no way it can

be 2 and 4, or 2 and 3…not at the same time. You need to

take another look.

Joe: That or the drummer got it wrong. They’re not always

the brightest in the band.…

Frank: Could it be something to do with it being ratios rather

than sizes? Like, I’m twice as old as my sister, but I’m also half

as old as my dad.…

Joe: And you’re half as good-looking as me. What are you

going on about? Those are both twos…we’re worried about

different numbers!

Frank: Yeah, but my dad is four times as old as my kid

sister—so, you could say their ages were 4:1, even though my

dad and I are 2:1. They’re both true; it’s just relative.

Jim: Relative to what, though?

Frank: That’s the thing—maybe we just need to choose one

thing for our ratios to be relative to and then stick with it.

Frank

you are here 4 95