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Pythagorean theorem

c. b. a. Pythagorean theorem. Introduction. Activity ” Pythagorean theorem” History Proof of the theorem Examples. Activity. 0. 0. 1. 2. 1. 3. 4. 2. 5. 3. 0. 1. 2. 3. 4. 5. 4. 5. Now , Take a square cardboard and a ruler.

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Pythagorean theorem

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  1. c b a Pythagoreantheorem

  2. Introduction • Activity • ” Pythagorean theorem” • History • Proof of the theorem • Examples

  3. Activity

  4. 0 0 1 2 1 3 4 2 5 3 0 1 2 3 4 5 4 5 Now , Take a square cardboard and a ruler.. 1. Cut the cardboard so that the height is 3cm and the base is 4cm. 5 cm 3 cm 2. Measure the length of the hypotenuse of the outgoing triangle. 4 cm

  5. a2 = b2 = c2 = 9 16 25 Let's try to determine the connection between the hypotenuse and the two edges : c = 5 a = 3 b = 4 so the relationship is found : a2 + b2 = c2

  6. Pythagorean theorem

  7. Hypotenuse - Edge facing the right angle Pisagor teoremi a c b If we take hypotenuse c for each right triangle and a and b on the other two sides : c2 = a2+ b2

  8. Historical Background

  9. Pythagorean theorem Pythagoras (BC. 580-500) Pythagoras was a greek philosopher who was involved in the development of mathematics, interested in astronomy, and helped in the emergence of music theory.

  10. The proof of Pythagorean theorem

  11. a b a b b a a b Let's say that XYZT is a square with an edge length of "a + b" Y X c c c c Z T Now the square has 4 same right triangle and Which is smaller than itself and whose edge length is "c"

  12. a + b a b A B X Y a c b c a + b c b c a T Z C a D b = 4 + c2 Area of ABCD Area of XYZT = (a + b) 2 = a 2 + 2ab + b 2 2ab + c 2 a2 + b2 = c2

  13. Examples

  14. Hipotenüs Example 1. Find the lenght of AC A 16 B C 12 Solution : AC2 = 122 + 162 (Pythagorean theorem) AC2 = 144 + 256 AC2 = 400 AC = 20

  15. Pythagorean theorem Implementations If you move 16 km west from where a car is located and then turn left and move 12 km south, how many km away from the starting point? 16km K 12km ?

  16. 16 km B A 12 km C Solution : AB = 16 BC = 12 AC2 = AB2 + BC2 AC2 = 162 + 122 AC2 = 400 AC = 20 The distance from the starting point of the car to the arrival point is 20 km.

  17. Ahmet is flying a kite at a distance of 160 meters from a tree. The lenght of the kite rope is 200 meters. Ahmet is 1.2 m in height and the kite is at the top of the tree. How many elevations do you have on your flying kite? 200 m ? 1.2 m 160 m

  18. A 200 m C B 1.2 m 160 m Çözüm : The figure ABC is a right triangle : AB = 200 BC = 160 AB2 = AC2 + BC2 (Pythagorean theorem) 2002 = AC2 + 1602 AC2 = 14400 AC = 120 So the height of the kite from the ground : = AC + Height of Ahmet = 120 + 1.2 = 121.2 m

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