The Pythagorean theorem

1. The Pythagorean theorem The Pythagorean theorem Demonstrate the Pythagorean Theorem Pythagorean Theorem Test Pythagorean Triples Question The Distance Formula Example Problem Test Yourself The Pythagorean theorem

2. 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. The Pythagorean theorem

3. The Pythagorean theorem 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." C 90° hypotenuse A B The Pythagorean theorem

4. 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 Pythagorean theorem

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

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

7. "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." The Pythagorean theorem

8. The Pythagorean theorem

9. 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. The Pythagorean theorem

10. Pythagorean Theorem Test http://win.matematicamente.it/test/test_pitagora.htmlhttp://www.crctlessons.com/Pythagorean-theorem-test.htmlhttp://www.mathsisfun.com/pythagoras.htm The Pythagorean theorem

11. Pythagorean Triples The Pythagorean theorem

12. 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 The Pythagorean theorem

13. 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. The Pythagorean theorem

14. 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. 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." The Pythagorean theorem

15. 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 The Pythagorean theorem

16. The Distance Formula The Pythagorean theorem

17. 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: The Pythagorean theorem

18. 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 . The Pythagorean theorem

19. If H1 and H2are the projection of P1 and P2on the x axis, the segments P1Q and H1H2 are opposite sides of a rectangle, so that But so Similarly, The Pythagorean theorem

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

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

22. Example Problem 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 : Given the points ( 1, -2 ) and ( -3, 5 ), find the distance between them The Pythagorean theorem

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 : 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 ). The Pythagorean theorem