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Head First 2D Geometry by Dawn Griffiths, Stray

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T
his row
+
this row
a pattern worth investigating
The lengths of the sides are linked by a pattern
Did you find it? For the four right triangles we tested, it seems like the square of
the length of the longest side is equal to the squares of the other two sides added
together.
Triangle 3-4-5 6-8-10 5-12-13 9-12-15
Shortest-side’s
length
Middle-side’s
length
Longest-side’s
length
Shortest-side
squared
Middle-side
squared
Longest-side
squared
Shortest-side
squared +
middle-side
squared
3
4
5
9
16
25
9 + 16
= 25
6
8
10
36
64
100
36 + 64
= 100
5
12
13
25
144
169
25 +
144
= 169
9
12
15
81
144
225
81 +
144
= 225
= this row!
Right. So—for these four right triangles,
you get this freaky pattern. Don’t you think
if you’re gonna use this to design skate jumps
you need something a bit more reliable? What if
these are the ONLY four triangles it works for?
It doesn’t even make sense. What have squares
got to do with triangles anyway?
Chapter 3
True. Can we really trust this pattern?
The numbers we’ve tested so far seem pretty conclusive,
but do we definitely know that this pattern is going to work
for all the possible right-angled jumps we might need?
And how do the squares relate to the triangles? Let’s
investigate this pattern in a more general way.
126
b
b
b
b
b
squar
e with
sides “c”.
square wi
th
sides “a” AND
a white square
wi
th sides “b”.
tr
iangle
l
(congruent).
Put 4 in her
e.
P
ut 4 in her
e.
Spin the triangles around,
but don’t f
lip them ov
er.
the pythagorean theorem
Geometry Investigation Magnets
Let’s experiment with a general right triangle. The sides of the triangle
can be a, b, and c—where c is the longest side. Below are two large
squares, each of which has side length a+b.
Can you arrange the gray triangles inside the squares so that in one box
you are left with a square with side length c” and in the other box you
are left with two squares—one with side length a” and one with side
length “b”? Make sure to use four triangles in each box.
What do you know about the white area left in each box? What does this
tell you about how the pattern you found might work for a right triangle
with sides a, b, and c?
a
c
A general r
ight
a b
a
c
a
c
a
c
a
c
Cr
eate a whi
te
b
a
All of these gray
triangles ar
e identica
a
b
c
a
c
a
b
c
a b
a
b
c
b
Cr
ea
te a whi
te
a
b
you are here 4 127
tr
iangle
investigation solution
Geometry Investigation Magnets Solution
Lets experiment with a general right triangle. The sides of the triangle can
be a, b, and c—where c is the longest side. Below are two large squares,
each of which has side length a+b.
Can you arrange the gray triangles inside the squares so that in one box you
are left with a square with side length “cand in the other box you are left
with two squares—one with side length “a and one with side length “b”?
Make sure to use four triangles in each box.
What do you know about the white area left in each box? What does this tell
you about how the pattern you found might work for a right triangle with
sides a, b, and c?
a b
a
b
c
A general r
ight
The gray triangles are all congruent,
so the gray area we’ve created in each
b
box must be equal. This means that
the leftover area in white must also
be equal.
a
For a right triangle with sides a,
b, and c, the square of c is equal
to the squares of a and b added
together.
c
c
c
c
T
he ar
ea of
this squar
e is c
2
because i
t has
sides of length
“c”.
a b
a
b
a
a
b
b
Ar
ea of this
square is b
2
.
Ar
ea ofthis
squar
e is a
2
because i
ts
side length
is “a”.
128 Chapter 3

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