The

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The
Pythagorean
Theorem
c
a
b
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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 right triangle because it contains a
right angle.
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The measure of a right angle is 90o
90o
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in the
The little square
angle tells you it is a
right angle.
90o
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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,
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and you square the lengths of the two sides that
make up the right angle,
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and add them together,
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you get the same number you would get by squaring
the other side.
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Is that correct?
?
?
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It is. And it is true for any right triangle.
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The two sides which come together in a right
angle are called
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The two sides which come together in a right
angle are called
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The two sides which come together in a right
angle are called
legs.
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The lengths of the legs are usually called a and
b.
a
b
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The side across from the right angle
is called the
hypotenuse.
a
b
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And the length of the hypotenuse
is usually labeled c.
c
a
b
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The relationship Pythagorus discovered is now
called The Pythagorean Theorem
c
a
b
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The Pythagorean Theorem says, given the right
triangle with legs a and b and hypotenuse c,
c
a
b
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then
c
a
b
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You can use The Pythagorean Theorem to solve many
kinds of 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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Using The Pythagorean Theorem,
48
482
362


c2
36
c
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Why?
Can you see that we have a right triangle?
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Which side is the hypotenuse?
Which sides are the legs?
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Then all we need to do is calculate
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And you end up 60 miles from where you started.
So, since c2 is 3600, c is
48
36
60
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Find the length of a diagonal of the rectangle
?
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Find the length of a diagonal of the rectangle
?
c
b 8
a 15
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Find the length of a diagonal of the rectangle
17
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Practice using The
Pythagorean Theorem to solve these right
triangles
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13
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(a)
(c)
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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.
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Check It Out! Example 2 Continued
Rewrite the question as a statement.
Find the distance from one corner of the field
to the opposite corner of the field.
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.
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.
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Check It Out! Example 2 Continued
You can use the Pythagorean Theorem to write an
equation.
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Check It Out! Example 2 Continued
a2 b2 c2
Use the Pythagorean Theorem.
332 1002 c2
Substitute for the known variables.
1089 10,000 c2
Evaluate the powers.
11,089 c2
Add.
105.304 ? c
Take the square roots of both sides.
105.3 ? c
Round.
The distance from one corner of the field to the
opposite corner is about 105.3 yards.
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The Pythagorean Theorem

• For any right triangle, the sum of the areas of
  the two small squares is equal to the area of the
  larger.
• a2 b2 c2

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Proof
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Lets look at it this way
a
c
a
c
b
b
c2
a2
b2
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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 the hypotenuse?
• Which sides are the legs?
• Now use a2 b2 c2

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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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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?

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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 c2 to solve for the missing side.

Distance from house 7 meters
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Ladder ProblemSolution

• 72 b2 252
• 49 b2 625
• b2 576
• b 24 m
• How did you do?