Pythagorean.ppt

8/10/2019 Pythagorean.ppt

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a

b

c

222cba


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Pythagorean Theorem Essential

Questions

How is the Pythagorean Theorem used to

identify side lengths?

When can the Pythagorean Theorem be

used to solve real life patterns?


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This is a right triangle:


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We call it a righttriangle

because it contains a

right angle.


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The measure of a right

angle is 90o

90o


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The little square

90o

in the

angle tells you it is a

right angle.


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About 2,500 years ago, a

Greek mathematician named

Pythagorus discovered a

special relationship between

the sides of right triangles.


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Pythagorus realized that if

you have a right triangle,

3

4

5


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and you square the lengths

of the two sides that make

up the right angle,

2423

3

4

5


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and add them together,

3

4

5

2423 22 43


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22 43

you get the same number

you would get by squaring

the other side.

222 543

3

4

5


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Is that correct?

222543

?

25169 ?


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It is. And it is true for any

right triangle.

8

6

10222 1086

1006436


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The two sides which

come together in a rightangle are called


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The two sides which



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The two sides which



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The side across from the

right angle

a

b

is called the


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And the length of the

hypotenuseis usually labeled c.

a

b

c


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The relationship Pythagorus

discovered is now called

The Pythagorean Theorem:

a

b

c


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The Pythagorean Theorem

says, given the right trianglewith legs aand band

hypotenuse c,

a

b

c


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then

a

b

c

.222

cba


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You can use The Pythagorean

Theorem to solve many kindsof problems.

Suppose you drive directly

west for 48 miles,

48


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Then turn south and drive for

36 miles.

48

36


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How far are you from where

you started?

48

36

?


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482

Using The Pythagorean

Theorem,

48

36

c

362+ =c2


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Why?

Can you see that we have a

right triangle?

48

36c

482 362+ =c2


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Which side is the hypotenuse?

Which sides are the legs?

48

36

c

482 362+ =c2


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22 3648

Then all we need to do is

calculate:

12962304

3600 2c


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And you end up 60 miles from

where you started.

48

36

60

So, since c2is 3600, cis 60.So, since c2is 3600, cis


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Find the length of a diagonal

of the rectangle:

15"

8"?


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of the rectangle:

15"

8"?

b = 8

a = 15

c


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of the rectangle:

15"

8"17


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Practice using

The Pythagorean Theoremto solve these right triangles:


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5

12

c=13


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10

b

26

24


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10

b

26

=24

(a)

(c)

222 cba

2222610 b

676100 2 b

1006762

b

5762b

24b


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Check It Out!Example 2

A rectangular field has a length of 100 yards and awidth 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.


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Check It Out!Example 2 Continued

1 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.


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2 Make a Plan

You can use the Pythagorean Theorem to

write an equation.


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Solve3

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.


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Proof

L t l k t it


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a

a

a2

bb

c

c

b2

c2

Lets look at it

this way


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Baseball Problem

A baseball diamond is really a square.

You can use the Pythagorean theorem to find

distances around a baseball diamond.


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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?


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Baseball Problem

To use the Pythagorean

theorem to solve for x,

find the right angle.

Which side is thehypotenuse?

Which sides are the legs?

Now use: a2+ b2= c2


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Baseball Problem

Solution The hypotenuseis 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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Ladder Problem

A ladder leans against a

second-story window of a

house.

If the ladder is 25 meterslong,

and the base of the ladder

is 7 meters from the

house,how high is the window?


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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= c2to solve

for the missing side.

Distance from house: 7 meters


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Ladder ProblemSolution

72+ b2= 252

49 + b2= 625b2= 576

b = 24 m

How did you do?A = 7 m

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Pythagorean.ppt