Head First 2D Geometry

pick a single starting point

Ratios need to be consistent

The ratios we took from the diagram are individually

correct, but they describe different relationships between

the lengths of our lines.

On our design, we need to make sure that we reflect all

the ratios at the same time, which means we have to pick

one thing and then work everything out relative to that.

4

3

2

1

W

e need t

o make

a

ll these ra

tios

w

ork t

ogether.

Let’s choose this

2

Now we need to

1

3

We’re ready

biggest arrow to

include the 4:3 ratio

between the large

to use ratios!

be the thing we’re

working relative to.

and small arrows.

4

4

Divide 4 by 2 to get the

correct ratio for the square.

2

3

Divide 3 by 2 to get the

3

1

correct ratio for the square.

2

1.5

96 Chapter 2

That decimal

looks like it could be

tricky; if we picked some

different numbers, could

we get rid of it?

While the decimal isn’t really a problem, it’s certainly

easier to work with ratios that are only whole numbers:

your brain can compare ratios like 3:4 and 7:8 in a way

that you probably can’t just figure out which is bigger

out of 2.67:5.3 and 4.56:6.2.

Use y

our answ

er

s from

similarity and congruence

The ratios have turned out to be 4, 3, 2, and 1.5.

What would be the smallest set of whole numbers you could

substitute and still keep the ratios the same?

Time to get etching! If the design will fit on Liz’s brother’s iPod with the biggest arrow

head edges at 2.4cm long, what lengths do the other lines—a, b, c—need to be?

here in this answ

er.

2.4cm

b

a

c

you are here 4 97

double it up

The ratios have turned out to be 4, 3, 2, and 1.5.

What would be the smallest set of whole numbers you could

substitute and still keep the ratios the same?

If we multiply all the ratios by 2, then we

get a set of whole numbers with the same

ratios: 8, 6, 4, and 3.

4

2

3

1.5

8

6

4

3

98 Chapter 2

Time to get etching! If you make the big arrowhead edges 2.4cm long, what lengths

do the other lines—a, b, c—need to be?

b

a

c

First we need to find the factor for scaling a, b, and c:

The length with a ratio of 8 is 2.4cm, so scale = 2.4cm = 0.3cm

8

Then multiply all the other ratios by 0.3: a = 6 x 0.3 = 1.8 cm

b = 4 x 0.3 = 1.2cm

c = 3 x 0.3 = 0.9cm

2.4cm