these rows.

v

a

lue gets rea

l close t

o the thing it’s nearl

y

equa

l t

o, but nev

er qui

te gets ther

e.

W

e a

lr

eady know ev

erything w

e need t

o,

bef

or

e w

e ev

en compar

e height and base,

but w

e found on page 167 that the height

w

as 75m, so tha

t a

lso fi

ts wi

th this row.

you’ve got the right speakers

The 60

o

speakers are spot on

We know that the sides of the field are longer than the base.

While this doesn’t allow us to say exactly what the apex angle

is—23º or 57º or something else—it does tell us that the apex

angle must be less than 60º. That means the 60º speakers

are ideal.

85m

85m

Field B

T

he side is longer

than the base...

80m

…so the tr

iangle is in

If apex angle is Sides and base Height and base

Almost 0º

Less than 60º

Exactly 60º

Between 60º and 90º

Exactly 90º

More than 90º

Almost 180º

Side ........................... base

Side ........................... base

Side ........................... base

Side ........................... base

Side ........................... base

Side ........................... base

Side ......................... 1/2 base

Height ......................... base

Height .........................1/2 base

Height .........................1/2 base

Height .........................1/2 base

Height .........................1/2 base

Height .........................1/2 base

Height ............................... 0

Greater than

Greater than

Equal to

Less than

Less than

Less than

Nearly equal to

Nearly equal to

Greater than

Greater than

Greater than

Equal to

Greater than

Less than

Math geek

s call this “approaching”—the

178 Chapter 4

triangle properties

Estimating Angles Up Close

b

s

s

h

T

his is the

base.

T

hese are the

sides—they’r

e

equa

l.

T

he a

ltitude

on the base

giv

es us the

height.

T

his angle,

opposi

te the

base, is the one

w

e’r

e es

tima

ting.

In an equil

a

tera

l

triangle y

ou can’t tell

the base f

rom the

sides…that’s OK!

Don’t try to use

this on a sca

lene

tr

iangle!

aº

T

he sides can never

be less than ha

lf

the base.

For an isoceles triangle you can use the relative values of

the base, the sides, and the height to find out roughly what

the apex angle (the one opposite the base) is.

Side

Height

Base

0º

180º

60º 90º

V

ery acute

triangles

Obtuse

triangles

Acute

triangles

Equila

tera

l

triangle

Right

triangle

Base

1/2 base

Plot sides and height against base to find which zone

your apex angle is in: obtuse, acute, or very acute.

you are here 4 179

$1000!

the festival is almost ready

$2000

Budget

T

ickets

$15,000

Fencing –$3,750

Speakers –$2,000

NET

$9,250

ROCK FESTIVAL

TO-DO LIST

1) Fi

nd a venue

2) Sor

t out secur

i

ty fenci

ng

3) Sound system!!

4) Dr

inks/me

rc

h s

t

and?

Grea

t choic

e ther

e—

y

ou sav

ed y

ourself

Q:

Are you sure it’s OK to be

estimating angles like this? Don’t we

always need to work out exactly what

the angle is?

A:If the problem or question requires

a precise answer along the lines of “what

is angle x,” then you’ll need to find the

angle exactly, but sometimes it’s possible

to solve a problem just by knowing roughly

what an angle is. For example, is it a right

angle? Or maybe just is it acute or obtuse?

Q:

And this is isosceles only, yeah?

A:Yes, this technique only works

reliably for an isoceles triangle—where

two angles and two sides are the same. An

equilateral triangle is an isoceles triangle

with an extra matching side, so it works

for those, too, but then they’re pretty easy

to spot.

Q:

What if I only know the base and

the side, or the base and the height?

A:Sometimes that’s all you need. Any

time the base is greater than twice the

height you know you’ve got an angle more

than 90º.You really only need both for that

tricky zone between 60º and 90º.

But don’t worry—if you’ve got two out of

three for the base, side, and height lengths,

you can use the good old Pythagorean

Theorem to find the other. (This only works

for right and isoceles triangles—don’t

try it on a scalene triangle or you’ll come

unglued.)

Q:

OK—but how do I remember this?

I’m bound to get mixed up about the

base and sides and and which is greater

and less than which! I’m not a computer

you know.…

A:It’s almost certainly easier to

remember the zones on the graph than it is

to remember the inequalities.

So, sketch that graph, mark what you know

about the 60º and 90º triangles on it, and

label the zones.

Q:

Really? You think I could draw it

just like that?

A:Go on…give it a shot on a scrap of

paper now. It’s a really time efficient way to

check your answers in an exam by the way.

180 Chapter 4

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