Pythagorean Pythagorean TheoremTheorem

Obj: SWBAT identify and apply the Obj: SWBAT identify and apply the Pythagorean Thm and its converse Pythagorean Thm and its converse to find missing sides and prove to find missing sides and prove triangles are righttriangles are right

Standard: M11.C.1.4.1 Find the Standard: M11.C.1.4.1 Find the measure of a side of a right measure of a side of a right triangle using the Pythagorean triangle using the Pythagorean ThmThm

History of the Pythagorean History of the Pythagorean ThmThm

At the height of their power, nearly a At the height of their power, nearly a millennium before Pythagoras, circa millennium before Pythagoras, circa 1900 - 1600 BCE , the Babylonians 1900 - 1600 BCE , the Babylonians (Babylon located in modern day Iraq) (Babylon located in modern day Iraq) identify what are now called identify what are now called Pythagorean triples (a set of positive Pythagorean triples (a set of positive integers a, b, c such that integers a, b, c such that aa2 2 + b+ b22 = c = c22

A Chinese astronomical A Chinese astronomical and mathematical treatise and mathematical treatise called the called the Chou Pei Suan Ching (The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven, ca. 500-200 B.C.), possibly predating Pythagoras, gives a statement of and geometrical demonstration of the Pythagorean Thm.

History of the Pythagorean History of the Pythagorean ThmThm

History of the Pythagorean History of the Pythagorean ThmThm

Despite evidence Despite evidence predating him, the predating him, the Greek named Greek named Pythagoras is credited Pythagoras is credited with the theorem. with the theorem. According to tradition, According to tradition, Pythagoras once said, Pythagoras once said, “Number rules the “Number rules the universe…” WHAT A universe…” WHAT A FREAKING GENIUS!!!!FREAKING GENIUS!!!!

leg

leg

hypotenuse

Right angle

Basics of the Right TriangleBasics of the Right Triangle

Pythagorean ThmPythagorean Thm In ANY In ANY rightright triangle, the triangle, the sumsum of the of the

squaressquares of the lengths of the of the lengths of the legslegs is is equalequal to the to the squaresquare of the length of of the length of the the hypotenusehypotenuse..

aa2 2 + b+ b22 = c = c22

Use the Pythagorean Thm to Use the Pythagorean Thm to solve for x.solve for x.

3

4

x

1. Square both 1. Square both legslegs

3 ft

4 ft

4ft

3 ft

1 3

4 5 6

7 8 9

1

2

2 3 4

5 6 7 8

9 10 11 12

13 14 15 16

2. Count the total 2. Count the total squaressquares

3 ft

4 ft

4ft

3 ft

1 3

4 5 6

7 8 9

1

2

2 3 4

5 6 7 8

9 10 11 12

13 14 15 16

9 + 16 = 25

3 ft

4 ft

4ft

3 ft

1 3

4 5 6

7 8 9

1

2

2 3 45 6 7 8

9 10 11 1213 14 15 16

9 + 16 = 25

3. Put that number of squares 3. Put that number of squares on the hypotenuseon the hypotenuse

2

4

6

8

10

12

14

16

18

20

22

24

1

3

5

7

9

11

13

15

17

19

21

23

25

3 ft

4 ft

4ft

3 ft

1 3

4 5 6

7 8 9

1

2

2 3 45 6 7 8

9 10 11 1213 14 15 16

9 + 16 = 25

4. Count the number of squares that touch the hypotenuse.

2

4

6

8

10

12

14

16

18

20

22

24

1

3

5

7

9

11

13

15

17

19

21

23

25

# = 5

3 ft

4 ft

4ft

3 ft

1 3

4 5 6

7 8 9

1

2

2 3 45 6 7 8

9 10 11 1213 14 15 16

9 + 16 = 25

5.That number is the length of the hypotenuse.

2

4

6

8

10

12

14

16

18

20

22

24

1

3

5

7

9

11

13

15

17

19

21

23

25

# = 5Length = 5

LOOK FAMILIAR?!

WOW, it’s actually the Pythagorean Thm! That’s so freaking cool!!!

Why are the wrong answers wrong???

Pythagorean TriplesPythagorean Triples Are short cuts! They are sets of 3 Are short cuts! They are sets of 3

whole numberswhole numbers (a, b, and c) that (a, b, and c) that satisfy the equation satisfy the equation aa2 2 + b+ b22 = c = c2 2

Most frequent examples:Most frequent examples: *** 3, 4, 5 (where a=3, b=4, c=5) ****** 3, 4, 5 (where a=3, b=4, c=5) *** 5, 12, 135, 12, 13 8, 15, 178, 15, 17 7, 24, 257, 24, 25

ANY scale or multiple of Pythagorean ANY scale or multiple of Pythagorean triples will work!!!triples will work!!!

Pythagorean Triple ExamplePythagorean Triple Example

Don’t be fooled by the disguise…

...it’s still thePythagorean Thm!!

Pythagorean Thm application Pythagorean Thm application with even more Geometry!!! with even more Geometry!!! It’s actually that much more It’s actually that much more

fun!!!fun!!!

HOLY SMOKES, LOOK AT ALL HOLY SMOKES, LOOK AT ALL OF THIS GEOMETRY!!! A-MAZ-OF THIS GEOMETRY!!! A-MAZ-

ING!!!!ING!!!!