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The Pythagorean TheoremThe Pythagorean Theorem
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
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
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
ProofProof
6
8
x
2 2 2a b c 2 2 26 8 x
236 64 x 2100 x
2100 x
Solve for x.
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
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
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
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
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.
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.
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?
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
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
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?
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
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
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