Head First 2D Geometry

Is this angle less

than

60

º

?

triangle properties

o

100m will do, but can you rent the 60 speaker?

The 60º speakers are way cheaper but are they going to do the job?

Will everybody in the festival be able to hear or will people at the

sides be straining to hear?

That all depends on the angle of your venue. The speakers project

sound in the shape of an isosceles triangle. Your venue is also an

isosceles triangle. If the venue angle is less than 60º, then it’s all good.

Because 60º are the narrowest speakers available, it doesn’t really

matter what that angle is, as long as it’s less than 60º. If it’s more than

60º, then you’ll need to find an extra $1,000 for the 90º speakers.

Field B

Uh, yeah. Did

you notice that the

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Check any you think are true and that will help you find the

angle you need (it can be more than one).

venue diagrams don’t

HAVE angles?

That’s true.

The only thing you know about the venue is the side

lengths and height (you worked out the perimeter and area

earlier, too, but they don’t tell you the angles either). Is that

enough to help you work out if the 60º speaker will do?

Sides and angles change together.

From sides you can find angles accurately.

From sides you can sometimes find angles accurately, and sometimes find them roughly.

Sides and angles change separately.

Sides tell you nothing about angles.

sides and heights help find angles

Check any you think are true and that will help you find the

angle you need (it can be more than one).

Sides and angles change together.

From sides you can find angles accurately.

From sides you can sometimes find angles accurately, and sometimes find them roughly.

Sides and angles change separately.

Sides tell you nothing about angles.

Solution

Sides and height tell you a lot about angles

Imagine a triangle with three sides equal length. Do you

know what the angles of that triangle would be?

The answer is upside down at the bottom of this page, but

you’ve probably already worked it out.…

It’s an equilateral triang le so the ang les are each 60º

Whatever…dude,

can we use the 60º

speakers or not?

Don’t w

orry—he’s

N

O

T taking the

s

tage a

t the f

es

tiva

l.

80m

85m

85m

Field B

Y

our venue, wi

th side

lengths but no angles.

176 Chapter 4

Y

ou could cut up

straws or pap

er

s

trips int

o equa

l

lengths and try

this wi

th those,

t

oo.

triangle properties

Geometry Detective

Your mission is to complete the table at the bottom of this page,

by using your own hands to investigate how we can use side

length and height to estimate angles.

Ap

ex

angle.

Move y

our thumbs t

o

change the base length.

Using your forefingers and thumbs, make a

triangle that is roughly equilateral.

Now change the length of the base of the

triangle, by moving your thumbs apart or

together. Keep your fingers the same length

so that you know the triangle is isosceles.

Pay attention to how changing the

proportions of the triangle—the relative

length of the base and sides, and the

height—changes the apex angle.

Complete the table by filling in each blank

using one of the following terms:

Greater than Less than

Equal to Nearly equal to

If apex angle is Sides and base Height and base

Almost 0º

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

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

Less than 60º

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

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

Exactly 60º

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

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

Between 60º and 90º

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

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

Exactly 90º

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

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

More than 90º

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

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

Almost 180º

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

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

Based on your table, do you think the 60º speaker will be OK?

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