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About 2,500 years ago, a Greek mathematician named Pythagoras discovered a special relationship between the sides of right triangles.
Pythagoras was not only a mathematician, but a philosopher and religious leader, as well.
He was responsible for many important developments in math,
astronomy, and
music.
Pythagoras (~560-480 B.C.)
His students formed a secret society called the Pythagoreans.
As well as studying math, they were a political and religious organization.
Members could be identified by a five pointed star they wore on their clothes.
The Secret Brotherhood
They had to follow some unusual rules. They were not allowed to wear wool, drink wine or pick up anything they had dropped!
The Secret Brotherhood
The initiation into the secret society asked that for the first 5 years into the brotherhood, they would not speak. Just listen.
A right angled triangle
A Pythagorean Puzzle
Draw a square on each side.
A Pythagorean Puzzle
cb
a
Measure the length of each side
A Pythagorean Puzzle
Work out the area of each square.
A Pythagorean Puzzle
a
b
C²
b²
a²
c
A Pythagorean Puzzle
c²
b²
a²
Proof
a
a
a2
bb
cc
b2
c2
Let’s look at it this way…
A Pythagorean Puzzle
1
A Pythagorean Puzzle
1
2
A Pythagorean Puzzle
1
2
A Pythagorean Puzzle
1
2
3
A Pythagorean Puzzle
1
2
3
A Pythagorean Puzzle
1
23
4 A Pythagorean Puzzle
1
23
4
A Pythagorean Puzzle
1
23
45
A Pythagorean Puzzle
1
23
4
5
What does this tell you about the areas of the three squares?
The red square and the yellow square together cover the green square exactly.
The square on the longest side is equal in area to the sum of the squares on the other two sides.
A Pythagorean Puzzle
1
23
4
5
Put the pieces back where they came from.
A Pythagorean Puzzle
1
23
45
A Pythagorean Puzzle
Put the pieces back where they came from.
1
23
4
5
A Pythagorean Puzzle
Put the pieces back where they came from.
1
2
3
4
5
A Pythagorean Puzzle
Put the pieces back where they came from.
1
23
4
5
A Pythagorean Puzzle
Put the pieces back where they came from.
1
23
4
5
A Pythagorean Puzzle
Put the pieces back where they came from.
This is the Pythagorean Theorem.
A Pythagorean Puzzle
c²
b²
a²
c²=a²+b²
It only works with right-angled triangles.
hypotenuse
The longest side, which is always opposite the right-angle, has a special name:
This is the name of Pythagoras’ most famous discovery.
Pythagorean Theorem
c
b
a
c²=a²+b²
Pythagorean Theorem
Pythagoras realized that if you have a right triangle,
3
4
5
and you square the lengths of the two sides that make up the right angle,
24233
4
5
and add them together,
3
4
5
2423 22 43
22 43
you get the same number you would get by squaring the other side.
222 543 3
4
5
This is true for any right triangle.
8
6
10222 1086
1006436
Practice using The Pythagorean Theorem to solve these right triangles:
5
12
c = 13
10
b
26
10
b
26
= 24
(a)
(c)
222 cba 222 2610 b
676100 2 b1006762 b
5762 b24b
You can use The Pythagorean Theorem to solve many kinds of problems.
Suppose you drive directly west for 48 miles,
48
Then turn south and drive for 36 miles.
48
36
How far are you from where you started?
48
36?
482
Using The Pythagorean Theorem,
48
36c
362+ = c2
Why? Can you see that we have a right triangle?
48
36c
482 362+ = c2
Which side is the hypotenuse? Which sides are the legs?
48
36c
482 362+ = c2
22 3648
Then all we need to do is calculate:
12962304
3600 2c
And you end up 60 miles from where you started.
48
3660
So, since c2 is 3600, c is 60.So, since c2 is 3600, c is
Find the length of a diagonal of the rectangle:
15"
8"?
Find the length of a diagonal of the rectangle:
15"
8"?
b = 8
a = 15
c
222 cba 222 815 c 264225 c 2892 c 17c
b = 8
a = 15
c
Find the length of a diagonal of the rectangle:
15"
8"17
Check It Out! Example 2
A rectangular field has a length of 100 yards and a width of 33 yards. About how far is it from one corner of the field to the opposite corner of the field? Round your answer to the nearest tenth.
Check It Out! Example 2 Continued
11 Understand the Problem
Rewrite the question as a statement.
• Find the distance from one corner of the field to the opposite corner of the field.
• The segment between the two corners is the hypotenuse.
• The sides of the fields are legs, and they are 33 yards long and 100 yards long.
List the important information:
• Drawing a segment from one corner of the field to the opposite corner of the field divides the field into two right triangles.
Check It Out! Example 2 Continued
22 Make a Plan
You can use the Pythagorean Theorem towrite an equation.
Check It Out! Example 2 Continued
Solve33
a2 + b2 = c2
332 + 1002 = c2
1089 + 10,000 = c2
11,089 = c2
105.304 c
The distance from one corner of the field to the opposite corner is about 105.3 yards.
Use the Pythagorean Theorem.
Substitute for the known variables.
Evaluate the powers.
Add.
Take the square roots of both sides.
105.3 c Round.
Ladder Problem
A ladder leans against a second-story window of a house. If the ladder is 25 meters long, and the base of the ladder is 7 meters from the house, how high is the window?
Ladder ProblemSolution
First draw a diagram that shows the sides of the right triangle.
Label the sides: Ladder is 25 m
Distance from house is 7 m
Use a2 + b2 = c2 to solve for the missing side. Distance from house: 7 meters
Ladder ProblemSolution
72 + b2 = 252
49 + b2 = 625
b2 = 576
b = 24 m
How did you do?
A = 7 m
Baseball Problem
A baseball “diamond” is really a square.
You can use the Pythagorean theorem to find distances around a baseball diamond.
Baseball Problem
The distance between
consecutive bases is 90
feet. How far does a
catcher have to throw
the ball from home
plate to second base?
Baseball Problem
To use the Pythagorean theorem to solve for x, find the right angle.
Which side is the hypotenuse?
Which sides are the legs?
Now use: aa22 + b + b22 = c = c22
Baseball ProblemSolution
The hypotenuse is the distance from home to second, or side x in the picture.
The legs are from home to first and from first to second.
Solution:
x2 = 902 + 902 = 16,200
x = 127.28 ft
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