Head First 2D Geometry

exercise solution

a

b

c

b

The dark triangle is turned 60º compared to the

light one, and each angle of the triangle is 60º so

these edges are parallel. A line which is perpendicular

to one line is also perpendicular to lines parallel to

that line, so b is also 90º (another 6).

a

c

To find angle c, draw a

line parallel to the base of

the dark triangle.

60º

This creates some equal angles—the Z pattern. Angle c is

this Z angle (60º) plus the perpendicular (it’s one side of

the square and we drew our line parallel to the bottom of

the square), so

c = 60º + 90º = 150º (6 of these, too!)

48 down, 12 t

o go....

54 down…jus

t six left…

a

150º

To find angle a, first find the angles of this triangle.

This angle is the supplement of a right angle we already found, so it’s 90º.

This angle is the supplement of the 150ºwe just found, so it’s 30º.

The angle next to a is the remaining angle in the triangle = 60º, so a is the

supplement of this angle = 180º- 60º = 120º (6 of these as well).

…continued

88 Chapter 2

similarity and congruence

Tonight’s talk: Similarity and Congruence

Similarity:

I’m worried about you. You seem neurotic, always

concerned about being a size zero, or seven, or

whatever. You need to relax more.

I just think if you stopped worrying so much about

size and focused on proportionality you’d have more

opportunities. I get used all the time—my flexibility

is a real asset.

Faster? How? I’m pretty fast, you know!

True. And I’m a big stickler for efficiency. People

would do well to use both of us more!

Oh, tell me about it! I think they forget that we’re

not all about the triangles. And that we travel so

well.…

Congruence:

But size DOES matter. It’s all very well having your

angles right on paper, but if something is too big or

too small, in the real world size actually matters.

And potentially a real headache! Sometimes it

doesn’t cut it just being similar—and that’s where

I come in, when you need to rely on size as well as

shape. Plus, I’m faster to use.

Yeah, you’re quick, but even if something’s similar,

if it’s not congruent, then you’ve gotta do some

math for the lengths. OK, it’s only multiplication

and division, but it all takes time.

Yes. Are you getting much non-triangle work these

days? I’m mostly getting triangle stuff still, and it’s

not that I don’t like it…but you know, I have so

much more potential.

Completely—I love to travel! You can flip me

upside down, back to front, spin me around and

move me from one place to another, and I still work

just as well.

Great to catch up! I’ll let you know if I get any work

you could help out with.

you are here 4 89

expanding your design

Hey—I had a cool idea. Is there any

way you could work up the design so

that maybe we could get some t-shirts

printed, too? The band would be psyched!

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.

It’s an interesting suggestion—could you represent the

design in a way that means that you could use it to etch

gadgets of any size, and maybe even print it on T-shirts?

Can y

ou r

eally get

this diagram t

o fi

t

p

er

f

ec

tl

y onto a

ll of

these wi

thout doing a

t

on of w

ork f

or each

dif

f

er

ent size?

90 Chapter 2

similarity and congruence

It would be cool to finish marking up your design of the band’s logo so that it would

fit on any size gadget—even a T-shirt.

But what does that actually mean you need to do?

a) Draw the design at different sizes and mark up different lengths on each of

them, that way you’ll always have one to fit.

b) Forget the final size, and just make sure you’ve captured the relationships

between the lengths of the lines, so it stays proportional.

Wr

i

te what y

ou’d

need t

o do and

why you do or

don’t think i

ts

the

bes

t approach.

I think you’re getting

a bit ahead of yourself

here—we don’t have any lengths

at all yet. It’s way too early to be

thinking about resizing it…

you are here 4 91