5.4 – Pythagorean Theorem

1. 5.4 – Pythagorean Theorem

2. GEO-METRY • “geometry” is Greek for “earth measure” • In ancient Egypt, the Nile flooded every year • Good for crops / Bad for property boundaries

3. GEO-METRY • The Egyptians had to devise a method for re-determining boundaries after each flood • Greek mathematicians later redeveloped these methods and called it ? • GEOMETRY!!!!!!!!!!!!!!!!!!!!!!

4. Pythagoras (569 BC – 475 BC)

5. Pythagorean Theorem • For any right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse • a2+b2=c2 c a b

6. Thanks, Mr. Pythagoras!

7. Proof (just one of the many) (A + B) ² = 4(½AB) + C² A² + 2AB + B² = 2AB + C² A² + B² = C²

8. Hypo what? • What is a hypotenuse? • A side of a right triangle • Where is the hypotenuse? • Always across from the 90° angle • It is always the biggest side

9. Can a right triangle walk? • Yes because it has • LEGS!!!!! • They are the other two sides of a right triangle • They are always smaller than the ? • hypotenuse

12. But is it even a triangle? • Triangle inequality theorem • The sum of the lengths of any two sides of a triangle is greater than the length of the third side

13. Is it a triangle? • Do the following lengths make a triangle? • 3, 3, 8 • 2, 5, 8 • 4, 5, 8 • 3, 4, 4

14. What kind of triangle is it? • C is the biggest side of the triangle • If c2 > a2+b2 then ΔABC is obtuse • If c2 < a2+b2 then ΔABC is acute • If c2 = a2+b2 then ΔABC is right

15. What kind of triangle is it? • Do the following three segments form a triangle? If so, what kind? • 4, 5, 8 • 3, 4, 5 • 2, 2, 8 • 4, 4, 5

16. What is a Pythagorean Triple? • A hit in baseball that gets a Greek player to third base? • 3 Greek siblings born on the same day? • 3 whole numbers that are sides of a right triangle?

17. Examples of Pythagorean Triples • 3, 4, 5 • Because 5² = 3² + 4² • 5, 12, 13 • Because 13² = 5² + 12²