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Head First 2D Geometry by Dawn Griffiths, Stray

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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
you are here 4 175
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?
you are here 4 177

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