The Pythagorean theorem. Demonstrate the Pythagorean Theorem Many different proofs exist for this most fundamental of all geometric theorems

1. The Pythagorean theorem
1. The Pythagorean theorem
2. Demonstrate the Pythagorean Theorem
3. Pythagorean Theorem Test
4. Pythagorean Triples
5. Question
6. The Distance Formula
7. Example Problem
8. Test Yourself
Marcello Pedone The Pythagorean theorem

2. 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.

3. Marcello Pedone The Pythagorean theorem
The Pythagorean theorem
A B
C
hypotenuse
90°
Right triangle "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."

4. Marcello Pedone The Pythagorean theorem
The Pythagorean theorem
1 2Q Q Q 
The sum of the areas of the two squares on the legs equals the area of
the square on the hypotenuse.
2 2 2
AB BC CA 

5. Marcello Pedone The Pythagorean theorem
1 2 3 4 5
6 7 8 9 10
11 12 13 14 15
16 17 18 19 20
21 22 23 24 25
9(32)
16(42)
25(52)
2 2 2
5 3 4 
25=9+16
Demonstrate the Pythagorean Theorem
Many different proofs exist for this most
fundamental of all geometric theorems
The sum of the areas of
the two squares on the
legs equals the area of
the square on the
hypotenuse.
B
C
A
hypotenuseRight triangle
90°

6. Marcello Pedone The Pythagorean theorem
1
1
2
2
3
3
4
4
5
5
B
C
A
2 1 2Q   1 2 3 4 5Q     
1 3 4 5Q   
1 2Q Q Q 
The sum of the areas
of the two squares
on the legs equals
the area of the
square on the
hypotenuse.
Several beautiful and
intuitive proofs by shearing
exist

7. Marcello Pedone 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."

8. Marcello Pedone The Pythagorean theorem

9. Marcello Pedone The Pythagorean theorem
The sum of the areas of the two squares on the legs equals the area of the
square on the hypotenuse.
The Indian mathematician Bhaskara
constructed a proof using the above
figure, and another beautiful dissection
proof is shown below

10. 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

11. Marcello Pedone The Pythagorean theorem
Pythagorean Triples

12. 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

13. Marcello Pedone The Pythagorean theorem
The formula that will generate all Pythagorean
triples first appeared in Book X of Euclid's
Elements:
where n and m are positive integers of opposite
parity and m>n.
2 2
2 2
2x m n
y m n
z m n
 
 
 

14. 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.
Since the Pythagorean Theorem is NOT true, this triangle is NOT a
right triangle.
?2 2 2
?
10 6 7 ;
100 36 49
100 85
 
 


15. Marcello Pedone The Pythagorean theorem
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
Question

16. Marcello Pedone The Pythagorean theorem
The Distance Formula

17. Marcello Pedone The Pythagorean theorem
The distance
between points P1 and P2 with coordinates (x1, y1) and
(x2,y2) in a given coordinate system is given by the
following distance formula:
   
2 2
1 2 1 2 1 2PP x x y y   
1 2PP

18. Marcello Pedone The Pythagorean theorem
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 .
     
2 2 2
1 2 1 2PP PQ PQ 

19. Marcello Pedone 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 ,
1 1 2PQ H H
But
so that
1 2 1 2H H x x 
so
1 1 2PQ x x 
Similarly,
2 1 2PQ y y 

20. Marcello Pedone The Pythagorean theorem
         
2 2 2 2 2
1 2 1 2 1 2 1 2 1 2PP x x y y x x y y       
Taking square roots, we obtain the distance formula:
   
2 2
1 2 1 2 1 2PP x x y y   
1 1 2PQ x x 
2 1 2PQ y y 
Hence
     
2 2 2
1 2 1 2PP PQ PQ 

21. Marcello Pedone The Pythagorean theorem
EXAMPLE
ABThe distance between points A(2,5) and B(5,9) is
       
2 2 2 2
5 2 9 5 3 4 9 16 25 5AB          

22. Example Problem
Given the points ( 1, -2 ) and ( -3, 5 ), find the distance between them
Marcello Pedone The Pythagorean theorem
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 :
   
2 2
1 2 1 2 1 2PP x x y y   

23. 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 :
Marcello Pedone The Pythagorean theorem
   
2 2
1 2 1 2 1 2PP x x y y   

24. Test Yourself
1. Find the distance between the points ( -1, +4 ) and (+2, -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.
3. 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