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The Pythagorean The Pythagorean Theorem Theorem

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Page 1: The Pythagorean Theorem. Pythagoras Lived in southern Italy during the sixth century B.C. Lived in southern Italy during the sixth century B.C. Considered

The Pythagorean TheoremThe Pythagorean Theorem

Page 2: The Pythagorean Theorem. Pythagoras Lived in southern Italy during the sixth century B.C. Lived in southern Italy during the sixth century B.C. Considered

PythagorasPythagoras Lived in southern Lived in southern

Italy during the sixth Italy during the sixth century B.C.century B.C.

Considered the first Considered the first true mathematiciantrue mathematician

Used mathematics Used mathematics as a means to as a means to understand the understand the natural worldnatural world

First to teach that First to teach that the earth was a the earth was a sphere that revolves sphere that revolves around the sunaround the sun

Page 3: The Pythagorean Theorem. Pythagoras Lived in southern Italy during the sixth century B.C. Lived in southern Italy during the sixth century B.C. Considered

Right TrianglesRight Triangles

Longest side is the Longest side is the hypotenuse,hypotenuse, side side cc (opposite the 90(opposite the 90oo angle)angle)

The other two sides The other two sides are the are the legs,legs, sides sides aa and and bb

Pythagoras developed Pythagoras developed a formula for finding a formula for finding the length of the sides the length of the sides of any of any rightright triangle triangle

Page 4: The Pythagorean Theorem. Pythagoras Lived in southern Italy during the sixth century B.C. Lived in southern Italy during the sixth century B.C. Considered

The Pythagorean TheoremThe Pythagorean Theorem

““For any right For any right triangle, the sum triangle, the sum of the areas of the of the areas of the two small squares two small squares is equal to the is equal to the area of the larger.”area of the larger.”

aa22 + b + b22 = c = c22

Page 5: The Pythagorean Theorem. Pythagoras Lived in southern Italy during the sixth century B.C. Lived in southern Italy during the sixth century B.C. Considered

ProofProof

Page 6: The Pythagorean Theorem. Pythagoras Lived in southern Italy during the sixth century B.C. Lived in southern Italy during the sixth century B.C. Considered

6

8

x

2 2 2a b c 2 2 26 8 x

236 64 x 2100 x

2100 x

Solve for x.

Page 7: The Pythagorean Theorem. Pythagoras Lived in southern Italy during the sixth century B.C. Lived in southern Italy during the sixth century B.C. Considered

4

7

y

2 2 2a b c Solve for y.

2 2 27 4 y 249 16 y 265 y

265 y

y 8.1

Page 8: The Pythagorean Theorem. Pythagoras Lived in southern Italy during the sixth century B.C. Lived in southern Italy during the sixth century B.C. Considered

6

t15

2 2 2a b c Solve for t.

2 2 2t 6 15 2t 36 225

36 36 2t 1892t 189t 189t 13.7

Page 9: The Pythagorean Theorem. Pythagoras Lived in southern Italy during the sixth century B.C. Lived in southern Italy during the sixth century B.C. Considered

To the nearest tenth of a foot, find the length of the diagonal of a rectangle with a width of 4 feet and a length of 10 feet.

4 ft.

10 ft.

2 2 2a b c 2 2 24 10 x

216 100 x 2116 x

2116 x

x 116

x

x 10.8

Page 10: The Pythagorean Theorem. Pythagoras Lived in southern Italy during the sixth century B.C. Lived in southern Italy during the sixth century B.C. Considered

20 miles

A car drives 20 miles due east and then 45 milesdue south. To the nearest hundredth of a mile, how far is the car from its starting point?

45 milesx

2 2 2a b c 2 2 220 45 x

2400 2025 x 22425 x

22425 x

x 2425x 49.24

Page 11: The Pythagorean Theorem. Pythagoras Lived in southern Italy during the sixth century B.C. Lived in southern Italy during the sixth century B.C. Considered

ApplicationsApplications The Pythagorean theorem has far-reaching The Pythagorean theorem has far-reaching

ramifications in other fields (such as the arts), as ramifications in other fields (such as the arts), as

well as practical applications. well as practical applications.

The theorem is invaluable when computing The theorem is invaluable when computing

distances between two points, such as in navigation distances between two points, such as in navigation

and land surveying.and land surveying.

Another important application is in the design of Another important application is in the design of

ramps. Ramp designs for handicap-accessible sites ramps. Ramp designs for handicap-accessible sites

and for skateboard parks are very much in demand. and for skateboard parks are very much in demand.

Page 12: The Pythagorean Theorem. Pythagoras Lived in southern Italy during the sixth century B.C. Lived in southern Italy during the sixth century B.C. Considered

Baseball ProblemBaseball Problem

A baseball “diamond” is really a A baseball “diamond” is really a square.square.

You can use the Pythagorean theorem You can use the Pythagorean theorem to find distances around a baseball to find distances around a baseball diamond.diamond.

Page 13: The Pythagorean Theorem. Pythagoras Lived in southern Italy during the sixth century B.C. Lived in southern Italy during the sixth century B.C. Considered

Baseball ProblemBaseball Problem

The distance between The distance between

consecutive bases is 90consecutive bases is 90

feet. How far does a feet. How far does a

catcher have to throwcatcher have to throw

the ball from home the ball from home

plate to second base?plate to second base?

Page 14: The Pythagorean Theorem. Pythagoras Lived in southern Italy during the sixth century B.C. Lived in southern Italy during the sixth century B.C. Considered

Baseball ProblemBaseball Problem

To use the Pythagorean To use the Pythagorean theorem to solve for x, theorem to solve for x, find the right angle. find the right angle.

Which side is the Which side is the hypotenuse?hypotenuse?

Which sides are the Which sides are the legs?legs?

Now use: Now use: aa22 + b + b22 = c = c22

Page 15: The Pythagorean Theorem. Pythagoras Lived in southern Italy during the sixth century B.C. Lived in southern Italy during the sixth century B.C. Considered

Baseball ProblemBaseball ProblemSolutionSolution

The The hypotenusehypotenuse is the is the distance from home to distance from home to second, or side x in the second, or side x in the picture.picture.

The The legs legs are from home are from home to first and from first to to first and from first to second.second.

Solution: Solution: xx2 2 == 90 9022 + 90 + 9022 = =

16,20016,200 x = 127.28 ftx = 127.28 ft

Page 16: The Pythagorean Theorem. Pythagoras Lived in southern Italy during the sixth century B.C. Lived in southern Italy during the sixth century B.C. Considered

Ladder ProblemLadder Problem

A ladder leans against A ladder leans against a second-story window a second-story window of a house. of a house. If the ladder is 25 If the ladder is 25 meters long, meters long, and the base of the and the base of the ladder is 7 meters ladder is 7 meters from the house, from the house, how high is the how high is the window?window?

Page 17: The Pythagorean Theorem. Pythagoras Lived in southern Italy during the sixth century B.C. Lived in southern Italy during the sixth century B.C. Considered

Ladder ProblemLadder ProblemSolutionSolution

First draw a diagram First draw a diagram that shows the sides that shows the sides of the right triangle.of the right triangle.

Label the sides:Label the sides: Ladder is Ladder is 25 m25 m Distance from house Distance from house

is is 7 m7 m Use Use aa22 + b + b22 = c = c22 to to

solve for the missing solve for the missing side.side. Distance from house: 7 meters

Page 18: The Pythagorean Theorem. Pythagoras Lived in southern Italy during the sixth century B.C. Lived in southern Italy during the sixth century B.C. Considered

Ladder ProblemLadder ProblemSolutionSolution

7722 + b + b22 = 25 = 2522

49 + b49 + b22 = 625 = 625 bb22 = 576 = 576 b = 24 m b = 24 m

How did you do?How did you do? A = 7 m

Page 19: The Pythagorean Theorem. Pythagoras Lived in southern Italy during the sixth century B.C. Lived in southern Italy during the sixth century B.C. Considered

SourcesSources

Great info on the Pythagorean theorem, Great info on the Pythagorean theorem, Pythagoras, and other math-related Pythagoras, and other math-related topics:topics:

The Baseball Problem

Pythagoras of Samos

Pythagoras Playground

Microsoft Encarta 2000Microsoft Encarta 2000

http://www.geom.umn.edu/~demo5337/Group3/bball.html
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.html
http://www.kyes-world.com/pythagor.htm

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