Teorema di pitagora ol (2)

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

1. The Pythagorean theorem 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

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 The sum of the areas of the two squares on the legs equals the area of the square on the hypotenuse.

5. Marcello Pedone The Pythagorean theorem 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 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( 3 2 ) 16 ( 4 2 ) 25( 5 2 ) 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 hypotenuse Right triangle 90°

6. Marcello Pedone The Pythagorean theorem 1 1 2 2 3 3 4 4 5 5 B C A 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.

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.

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 P 1 and P 2 with coordinates (x 1 , y 1 ) and (x 2 ,y 2 ) in a given coordinate system is given by the following distance formula:

18. Marcello Pedone The Pythagorean theorem To see this, let Q be the point where the vertical line trough P 2 intersects the horizontal line trough P 1 . The x coordinate of Q is x 2 , the same as that of P 2 . The y coordinate of Q is y 1 , the same as that of P 1 . By the Pythagorean theorem .

19. Marcello Pedone The Pythagorean theorem If H 1 and H 2 are the projection of P 1 and P 2 on the x axis, the segments P 1 Q and H 1 H 2 are opposite sides of a rectangle , But so that so Similarly,

20. Marcello Pedone The Pythagorean theorem Taking square roots, we obtain the distance formula: Hence

21. Marcello Pedone The Pythagorean theorem EXAMPLE The distance between points A(2,5) and B(5, 9) is

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 ( x 1 , y 1 ) = ( -1, -2 ) and ( x 2 , y 2 ) = ( -3, 5 ) . Therefore, x 1 = -1 , y 1 = -2 , x 2 = -3 , and y 2 = 5 . To find the distance d between the points, use the distance formula :

23. Label the points as follows ( x 1 , y 1 ) = ( -1, -2 ) and ( x 2 , y 2 ) = ( -3, 5 ) . Therefore, x 1 = -1 , y 1 = -2 , x 2 = -3 , and y 2 = 5 . To find the distance d between the points, use the distance formula : Marcello Pedone The Pythagorean theorem

24. Test Yourself 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