c
a
b
About 2,500 years ago, a
Greek mathematician named
Pythagoras discovered a
special relationship between
the sides of right triangles.
Pythagoras (~560-480 B.C.)
Pythagoras was not only a mathematician, but a
philosopher and religious leader, as well.
He was responsible for many important
developments in math,
astronomy,
and music.
The Secret Brotherhood
His students formed a secret society called the
Pythagoreans.
As well as studying math, they were a political and
religious organization.
Members could be identified
by a five pointed star they
wore on their clothes.
The Secret Brotherhood
They had to follow some unusual rules. They were not
allowed to wear wool, drink wine or pick up anything
they had dropped!
The initiation into the secret society asked that for
the first 5 years into the brotherhood, they would
not speak. Just listen.
A Pythagorean Puzzle
A right angled triangle
A Pythagorean Puzzle
Draw a square on
each side.
A Pythagorean Puzzle
Measure the length
of each side
b
c
a
A Pythagorean Puzzle
Work out the area of
each square.
b²
b
C²
c
a
a²
A Pythagorean Puzzle
c²
b²
a²
Proof
Let’s look at it
this way…
a
a
b
c2
c
c
b
a2
b2
A Pythagorean Puzzle
A Pythagorean Puzzle

1
A Pythagorean Puzzle
1
2

A Pythagorean Puzzle
1
2
A Pythagorean Puzzle
1
2
3
A Pythagorean Puzzle
1
2
3
A Pythagorean Puzzle
1
3
2
4
A Pythagorean Puzzle
1
3
2
4
A Pythagorean Puzzle
1
3
2
5
4
A Pythagorean Puzzle
What does this
tell you about the
areas of the
three squares?
1
2
5
3
4
The red square and the yellow square
together cover the green square exactly.
The square on the longest side is equal in area to
the sum of the squares on the other two sides.
A Pythagorean Puzzle
Put the pieces back
where they came
from.
2
1
5
4
3
A Pythagorean Puzzle
Put the pieces back
where they came
from.
2
5
1
3
4
A Pythagorean Puzzle
Put the pieces back
where they came
from.
2
1
3
5
4
A Pythagorean Puzzle
Put the pieces back
where they came
from.
2
1
5
4
3
A Pythagorean Puzzle
Put the pieces back
where they came
from.
1
5
4
3
2
A Pythagorean Puzzle
Put the pieces back
where they came
from.
5
4
2
3
1
A Pythagorean Puzzle
c²
b²
This is the
Pythagorean
Theorem.
c²=a²+b²
a²
Pythagorean Theorem
This is the name of Pythagoras’ most
famous discovery.
It only works with right-angled triangles.
The longest side, which is always
opposite the right-angle, has a special
name:
Pythagorean Theorem
c
a
b
c²=a²+b²
Pythagoras realized that if
you have a right triangle,
5
3
4
and you square the lengths
of the two sides that make
up the right angle,
5
3
4
3
2
4
2
and add them together,
5
3
3 4
2
4
2
you get the same number
you would get by squaring
the other side.
5
3
3 4 5
2
4
2
2
This is true for any right
triangle.
6  8  10
2
2
2
10
8
36  64  100
6
Practice using
The Pythagorean Theorem
to solve these right triangles:
c = 13
5
12
b
10
26
b = 24
a b c
2
2
2
10  b  26
2
100  b  676
2
b  676  100
2
b  576
2
2
2
10 (a)
26
(c)
b  24
You
Suppose
can use
youThe
drive
Pythagorean
directly
Theorem
west for 48
to miles,
solve many kinds
of problems.
48
Then turn south and drive for
36 miles.
48
36
How far are you from where
you started?
48
36
?
Using The Pythagorean
Theorem,
2
2
48 + 36 = c
2
36
48
c
Why?
Can
you see that we have a
right triangle?
2
2
48 + 36 = c
2
36
48
c
Which sides
side isare
thethe
hypotenuse?
legs?
2
2
48 + 36 = c
2
36
48
c
Then all we need to do is
calculate:
48  36  2304 1296 
2
2
3600  c
2
2
Andsince
So,
you end
c isup
3600,
60 miles
c is 60.
from
where you started.
48
36
60
Find the length of a diagonal
of the rectangle:
15"
?
8"
Find the length of a diagonal
of the rectangle:
15"
b=8
c
?
a = 15
8"
15
225
acc 
17
8b
64
289 c
2
b=8
c
a = 15
2
2
Find the length of a diagonal
of the rectangle:
15"
17
8"
Check It Out! Example 2
A rectangular field has a length of 100 yards and a
width of 33 yards. About how far is it from one corner
of the field to the opposite corner of the field? Round
your answer to the nearest tenth.
Check It Out! Example 2 Continued
1
Understand the Problem
Rewrite the question as a statement.
• Find the distance from one corner of the field to the
opposite corner of the field.
List the important information:
• Drawing a segment from one corner of the field to the
opposite corner of the field divides the field into two
right triangles.
• The segment between the two corners is
the hypotenuse.
• The sides of the fields are legs, and they are 33 yards long
and 100 yards long.
Check It Out! Example 2 Continued
2
Make a Plan
You can use the Pythagorean Theorem to
write an equation.
Check It Out! Example 2 Continued
3
Solve
a2 + b2 = c2
332 + 1002 = c2
1089 + 10,000 = c2
11,089 = c2
105.304  c
105.3  c
Use the Pythagorean Theorem.
Substitute for the known variables.
Evaluate the powers.
Add.
Take the square roots of both sides.
Round.
The distance from one corner of the field to the
opposite corner is about 105.3 yards.
Ladder Problem
A ladder leans against a
second-story window of a
house.
If the ladder is 25 meters
long,
and the base of the ladder
is 7 meters from the
house,
how high is the window?
Ladder Problem
Solution
First draw a diagram that
shows the sides of the
right triangle.
Label the sides:
Ladder is 25 m
Distance from house is
7m
Use a2 + b2 = c2 to solve
for the missing side. Distance from house: 7 meters
Ladder Problem
Solution
72 + b2 = 252
49 + b2 = 625
b2 = 576
b = 24 m
How did you do?
A=7m
Baseball Problem
A baseball “diamond” is really a square.
You can use the Pythagorean theorem to find distances
around a baseball diamond.
Baseball Problem
The distance between
consecutive bases is 90
feet. How far does a
catcher have to throw
the ball from home
plate to second base?
Baseball Problem
To use the Pythagorean
theorem to solve for x,
find the right angle.
Which side is the hypotenuse?
Which sides are the legs?
Now use: a2 + b2 = c2
Baseball Problem
Solution
The hypotenuse is the
distance from home to
second, or side x in the
picture.
The legs are from home to
first and from first to
second.
Solution:
x2 = 902 + 902 = 16,200
x = 127.28 ft