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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