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Head First 2D Geometry by Dawn Griffiths, Stray

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Each of these
rows contains
the number
2, so tha
t's a
common f
ac
tor.
All the lengths are
H
ALF the size of
the 10-8-6 ramp.
number
s tell
y
ou wha
t the
lengths of sides
on y
our sca
led
ramp will be.
10
6
the pythagorean theorem
Any good jump has some similar scaled cousins
When you scale the ramp—making it bigger or smaller—none
of the angles change, so it stays a good ramp with a right angle.
To find whole number lengths for the smaller-scaled sizes, look
for a common factor in your current lengths. That’s easiest to do
by using a set of factor trees.
8
Use factor trees to find whole-number miniatures
Factor Tree Factor Tree Factor Tree
10 = 2 x 5
10 8 6
2 5 2 4 2 3
T
he other
5
3
4
What’s the next largest ramp you can build that is similar to the
10-8-6 design?
you are here 4 121
sharpen solution
What’s the next largest ramp you can build that is similar to the
10-8-6 design?
The lengths 15, 12, and 9 have the same ratios as
10-8-6 and as 5-4-3.
5 x 3 = 15
4 x 3 = 12
3 x 3 = 9
10 x 1.5 = 15
8 x 1.5 = 12
6 x 1.5 = 9
It doesn’t ma
tter whether y
ou
used the or
iginal ramp or the
mini ver
sion to do y
our sca
ling—
i
t w
orks out jus
t the same!
15
12
9
Q:
What if I wanted a ramp even smaller? Can I just keep
doing more factor trees?
A:The Kwik-klik units don’t come in half sizes—there isn’t a 1.5
length, so you’d quickly run out of parts, but assuming you weren’t
just talking about building it with the kit parts, you still only need
to do your factor trees until one of the numbers on the bottom is a
prime number—that means it can’t be divided by anything except
itself and one. Then to make a really small ramp you’d multiply
those factors by a fraction.
Q:
How can we skate on a 2D ramp? Isn’t this gonna be
more like a rail that you can slide on?
A:What we’re actually representing is the side of the ramp.
There would be two of these side triangles the same, with a panel
connected to the sloping beam on each triangle. This is a 3D
problem which has a 2D solution.
If you’re interested in exploring 3D problems further, come and
catch up with Sam in Head First 3D Geometry.
122 Chapter 3
W
ha
t the jumps
ac
tua
ll
y look like
in 3D
the pythagorean theorem
Kwik-Klik tips for easy right angles
Select lengths so that:
Longest side
2
= shortest side
2
+ middle side
2
Longest side
Middle side
Shortest
side
So how do we know which sets of ratios
can give us a right triangle? The dude at the
store gave me this slip of paper with some
odd stuff on it—tips for building right angles
or something.... I didn’t really pay it any
attention—what do you think it means?
you are here 4 123
study hall conversation
Kwik-Klik tips for easy right angles
Select lengths so that:
Longest side
2
= shortest side
2
+ middle side
2
Longest side
Middle side
Shortest
side
Well, it looks like
complete gibberish to me.
No wonder she didn’t pay
attention to it.
Frank: But it must mean something. I mean, nobody goes to
the trouble of writing something down unless it’s useful.
Jim: True. But how would you use it? And why would
squaring the side lengths have anything to do with the angle of
the triangle?
Joe: Oh…those twos are for squaring! Yeah—no way that
would work.
Frank: OK, don’t freak out, but if you just try it, like for the
3-4-5 ramp design…it works out perfect.
Jim: What? Are you sure you got your numbers right?
Frank: Yeah—I’m sure. Longest side is 5, and 5 squared is 25.
Shortest side is 3, and 3 squared is 9, and the middle side is 4,
and 4 squared is 16. So—add up the middle and shortest sides
squared—9 plus 16—and you get.…
Joe: 25. The same as the square of the longest side. That has
got to be a coincidence.
Frank: There’s only one way to find out—let’s check the
others.…
Frank
Jim
Joe
124 Chapter 3

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