The Pythagorean theorem
1.
2.
3.
4.
5.
6.
7.
8.
The Pythagorean theorem
Demonstrate the Pythagorean Theorem
Pythagorean Theorem Test
Pythagorean Triples
Question
The Distance Formula
Example Problem
Test Yourself
Marcello Pedone
The Pythagorean theorem
The Pythagorean theorem
Although Pythagoras is
credited with the famous
theorem, it is likely that the
Babylonians knew the result
for certain specific triangles
at least a millennium earlier
than Pythagoras. It is not
known how the Greeks
originally demonstrated the
proof of the Pythagorean
Theorem.
Marcello Pedone
The Pythagorean theorem
The Pythagorean theorem
Right triangle
C
"In any right triangle, the square
of the length of the hypotenuse
is equal to the sum of the
squares of the lengths of the
legs."
90°
hypotenuse
A
Marcello Pedone
B
The Pythagorean theorem
The Pythagorean theorem
The sum of the areas of the two squares on the legs equals the area of
the square on the hypotenuse.
Q  Q1  Q2
2
2
2
AB  BC  CA
Marcello Pedone
The Pythagorean theorem
Demonstrate the Pythagorean Theorem
16(42)
Many different proofs exist for this most
fundamental of all geometric theorems
9(32)
C
90°
25=9+16
hypotenuse
Right triangle
A
1
2
6
7
3
8
4
5
9
10
11 12 13 14 15
25(52)
Marcello Pedone
16 17 18 19 20
21
22 23theorem
24 25
The Pythagorean
B
52  32  42
The sum of the areas of
the two squares on the
legs equals the area of
the square on the
hypotenuse.
Q2  1  2 Q  1  2  3  4  5
3
Q1  3  4  5
4
C
1
Q  Q1  Q2
The sum of the areas
of the two squares
on the legs equals
the area of the
square on the
hypotenuse.
Marcello Pedone
Several beautiful and
intuitive proofs by shearing
exist
5
2
B
A
5
3
4
1
2
The Pythagorean theorem
"The area of the square built upon the
hypotenuse of a right triangle is equal
to the sum of the areas of the squares
upon the remaining sides."
Marcello Pedone
The Pythagorean theorem
Marcello Pedone
The Pythagorean theorem
The Indian mathematician Bhaskara
constructed a proof using the above
figure, and another beautiful dissection
proof is shown below
The sum of the areas of the two squares on the legs equals the area of the
square on the hypotenuse.
Marcello Pedone
The Pythagorean theorem
Pythagorean Theorem Test
http://win.matematicamente.it/test/test_pitagora.html
http://www.crctlessons.com/Pythagorean-theorem-test.html
http://www.mathsisfun.com/pythagoras.htm
Marcello Pedone
The Pythagorean theorem
Pythagorean Triples
Marcello Pedone
The Pythagorean theorem
Pythagorean Triples
There are certain sets of numbers that have a very special property. Not
only do these numbers satisfy the Pythagorean Theorem, but any
multiples of these numbers also satisfy the Pythagorean Theorem.
For example: the numbers 3, 4, and 5 satisfy the Pythagorean
Theorem. If you multiply all three numbers by 2 (6, 8, and 10), these new
numbers ALSO satisfy the Pythagorean theorem.
The special sets of numbers that possess this property are called
Pythagorean Triples.
The most common Pythagorean Triples are:
3, 4, 5
5, 12, 13
8, 15, 17
Marcello Pedone
The Pythagorean theorem
The formula that will generate all Pythagorean
triples first appeared in Book X of Euclid's
Elements:
x  2m  n
y  m2  n2
z  m2  n2
where n and m are positive integers of opposite
parity and m>n.
Marcello Pedone
The Pythagorean theorem
The Pythagorean theorem
"In any right triangle, the square of the length of the hypotenuse is equal to the sum of
the squares of the lengths of the legs."
A triangle has sides 6, 7 and 10.
Is it a right triangle?
The longest side MUST be the hypotenuse, so c = 10.
Now, check to see if the Pythagorean Theorem is true.
?
102 
 62  7 2 ;
?
100 
 36  49
100  85
Since the Pythagorean Theorem is NOT true, this triangle is NOT a
right triangle.
Marcello Pedone
The Pythagorean theorem
Question
1)If {x, 40, 41} is a Pythagorean triple, what is the value of x?
A: x = 9
B:x = 10
C:x = 11
D: x = 12
2) Which one of the following is not a Pythagorean triple?
A: 18, 24, 30
B:16, 24, 29
C:10, 24, 26
D:7, 24, 25
Marcello Pedone
The Pythagorean theorem
The Distance Formula
Marcello Pedone
The Pythagorean theorem
The distance PP
1 2
between points P1 and P2 with coordinates (x1, y1) and
(x2,y2) in a given coordinate system is given by the
following distance formula:
PP
1 2 
Marcello Pedone
The Pythagorean theorem
 x1  x2    y1  y2 
2
2
To see this, let Q be the point where the
vertical line trough P2 intersects the
horizontal line trough P1.
• The x coordinate of Q is
x2 , the same as that of
P2.
• The y coordinate of Q is
y1 , the same as that of
P1.
• By the Pythagorean
theorem .
 PP    PQ   P Q
2
1 2
2
1
Marcello Pedone
2
2
The Pythagorean theorem
If H1 and H2 are the projection of P1 and P2 on the
x axis, the segments P1Q and H1H2 are opposite
sides of a rectangle ,
so that
PQ
 H1H2
1
But
H1H 2  x1  x2
so
PQ
 x1  x2
1
Similarly,
P2Q  y1  y2
Marcello Pedone
The Pythagorean theorem
 PP    PQ   P Q
2
2
1 2
1
2
2
Hence
 PP    x  x    y  y 
2
1 2
2
1
2
1
2
2
  x1  x2    y1  y2 
2
2
Taking square roots, we obtain the distance formula:
PP
1 2 
 x1  x2    y1  y2 
2
2
P2Q  y1  y2
PQ
 x1  x2
1
Marcello Pedone
The Pythagorean theorem
EXAMPLE
The distance
AB 
AB
between points A(2,5) and B(5,9) is
 5  2    9  5
Marcello Pedone
2
2

3   4
2
2
 9  16  25  5
The Pythagorean theorem
Example Problem
Given the points ( 1, -2 ) and ( -3, 5 ), find the distance between them
Label the points as follows
( x1, y1 ) = ( -1, -2 ) and ( x2, y2 ) = ( -3, 5 ).
Therefore, x1 = -1, y1 = -2, x2 = -3, and y2 = 5. To find the distance d
between the points, use the distance formula :
PP
1 2 
Marcello Pedone
 x1  x2    y1  y2 
2
The Pythagorean theorem
2
Label the points as follows
( x1, y1 ) = ( -1, -2 ) and ( x2, y2 ) = ( -3, 5 ).
Therefore, x1 = -1, y1 = -2, x2 = -3, and y2 = 5. To find the distance d
between the points, use the distance formula :
PP
1 2 
Marcello Pedone
 x1  x2    y1  y2 
2
The Pythagorean theorem
2
Test Yourself
1.
2.
3.
Find the distance between the points ( -1, +4 ) and (+2, -2 ).
Given the points A and B where A is at coordinates (3, -4 ) and B is at coordinates ( -2, -8 )
on the line segment AB, find the length of AB.
Find the length of the line segment AB where point A is at ( 0,3 ) and point B is at ( -2, - 5 ).
Marcello Pedone
The Pythagorean theorem