The Pythagorean Theorem

1. The Pythagorean Theorem

2. The History of the Pythagorean Theorem • The Pythagorean Theorem takes its name from the ancient Greek mathematician Pythagoras (569 B.C.?-500 B.C.?), who was perhaps the first to offer a proof of the theorem. But people had noticed the special relationship between the sides of a right triangle long before Pythagoras.

3. The History of the Pythagorean Theorem • There is evidence that the Babylonians, more than 1000 years before Pythagoras, understood the relationship between the sides of a right triangle. There is also evidence that ancient Chinese philosophers understood the relationship between the sides of the right triangle around the same time as Pythagoras.

4. The History of the Pythagorean Theorem • Pythagoras himself was not simply a mathematician. He was an important philosopher who believed that the world was ruled by harmony and that numerical relationships could best express this harmony. He was the first, for example, to represent musical harmonies as simple ratios.

5. The History of the Pythagorean Theorem • We do not know for sure how Pythagoras himself proved the theorem that bears his name because he refused to allow his teachings to be recorded in writing. But probably, like most ancient proofs of the Pythagorean Theorem, it was geometrical in nature. • Today, we will prove the Pythagorean Theorem!

6. Right Triangles Only! • First, the Pythagorean Theorem only works in right triangles.  

7. Right Triangles Only! • The longest side of a right triangle that is opposite the right angle (the largest angle) is called the hypotenuse and usually labeled side “c”. • The two sides that are adjacent to the right angle are called the legs and are labeled sides “a” and “b” hypotenuse a c b legs

8. The Pythagorean Theorem • The Pythagorean Theorem states that the sum of the square of the legs of a right triangle is equal to the square of the hypotenuse.

9. The Pythagorean Theorem

10. The Pythagorean Theorem

11. The proof! +=…9 +16=25

12. Practice! • If a = 6 and b = 8, find c. • If a = 8cm and b = 15cm, find c.

13. Practice! • If c = 13 and b = 12, find a. • If c = 26m and a = 24m, find b.

14. Real World Applications! • The Pythagorean Theorem has many real-world applications. • It commonly used in the day to day life of • Engineers • Architects • Construction Workers • Criminal Forensic Scientists • NASA Scientists

15. Real World Applications! • The theorem can be used to figure out • the height of a wall • the breadth of a river • the depth of a lake • the trajectory of a bullet • the location of an airplane • GPS (Global Positioning System)

16. Real World Applications! • Advancements in technology have enabled computers to calculate many things that relate to the Pythagorean Theorem such as location. Google Earth can calculate the exact location of any places using a process called triangulation.

17. Real World Applications! • Triangulation is a method used for pinpointing a location with two known reference points. When Triangulation is used with a 90-degree angle, the Pythagorean Theorem is used to determine the location. More advanced levels of triangulation are based on Trigonometric Laws that apply to all triangles, not just right triangles.

18. The Pythagorean Theorem Baseball. • You've just picked up a ground ball at first base, and you see the other team's player running towards third base. How far do you have to throw the ball to get it from first base to third base, and throw the runner out?

19. The Pythagorean Theorem and the lock out. • You're locked out of your house and the only open window is on the second floor, 24 feet above the ground. You need to borrow a ladder from one of your neighbors. There's a bush along the edge of the house, so you'll have to place the ladder 10 feet from the house. What length of ladder do you need to reach the window?

20. 12 feet The Pythagorean Theorem and the Skateboard Park. 7 feet ??? • The city has decided to construct a new skateboard ramp for the skateboard park. The space available will allow a ramp to be 25 feet deep, 7 feet tall and span 5 feet across. To the nearest foot, how long will the ramp be? 25 feet

21. The Pythagorean Theorem and the Flagpole. • The line attached to the top of a flagpole is 100 feet long. When the free end of the line is placed on the ground, it is 60 feet from the base of the flag pole. How high is the flag pole?