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

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 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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