9.1

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9.1 9.2 The Pythagorean Theorem Its Converse

• HW Lesson 9.1 / 1-16
• Lesson 9.2/1-16

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Essential Understanding

• Use the the Pythagorean Theorem to solve
  problems.
• Use the Converse of the Pythagorean Theorem to
  solve problems.
• Use side lengths to classify triangles by their
  angle measures.

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Pythagorean Theorem
If You Have A Right Triangle, Then c²a² b²
c
a
b
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History of the theorem
Pythagoras of Samos was a Greek philosopher
responsible for many important developments in
mathematics!
But rumour has it Pythagoras Theorem was known
to the Babylonians some 1000 years before
Pythagoras.
However we all believe he was the first person
to prove the theorem and that is why the theorem
takes his name.
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Euclid
Pythagoras
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a2 b2c2
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The Pythagorean Theorem as some students see it.
a2b2c2
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A better way
c2
c
a2
a
a2b2c2
b
b2
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PYTHAGOREAN THEOREM
Applies to Right Triangles Only!
hypotenuse
c
leg
a
b
leg
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Find the missing side of the right triangle in
the 1 centimeter grid below.
x
6
8
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Find the missing side of the right triangle in
the 1 centimeter grid below.
12
5
x
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Find the missing side of the right triangle in
the 1 centimeter grid below.
x
4
7
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Find the length of the diagonal for a
rectangle that measures 3 inches by 4 inches.
x
3 in.
4 in.
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Find the Hypotenuse

• To find the hypotenuse, solve for c.
• 1)

• 2) a 3m, b 4m, find c.

 
 
 
 
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Find a leg

• You will not always solve for the hypotenuse (c).
  Sometimes you will have to find a leg (a or b).
• Example

 
 
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Find a leg
To find a leg, solve for a or b.

• 1)

• 2) b 30ft, c 34ft

 
13m
12m
 
 
b
 
b 5 m
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Pythagorean Theorem
 
 
17.03 miles c
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Pythagoras Questions
Pythagorean triple
Pythagorean triple
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Pythagoras Questions
 
x 6.32 m
 
x 21.11 cm
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d 11.07 cm
x 6.51 cm
?Perimeter 2(6.54.3) 21.6 cm
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The Converse Of The Pythagorean Theorem
If c² a² b², Then You Have A Right Triangle
c
a
b
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Using the Converse

• The Converse of the Pythagorean
• Theorem is True.
• Remember Converse means Reverse.

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Converse of the Pythagorean Theorem

• If c2 a2 b2 , then the triangle with sides a,
  b, and c is a right triangle.
• If a, b, and c are integers that satisfy the
  equation c2 a2 b2 , then a,
  b, and c are known as Pythagorean triples.

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Do These Lengths Form Right Triangles ?
5, 6, 10 6, 8, 10
10² __5² 6² 100___25 36 100? 61 NO
10²___6² 8² 100___36 64 100 100
YES
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Example of the Converse

• Determine whether a triangle with lengths 7, 11,
  and 12 form a right triangle.
• The hypotenuse is the longest length.

This is not a right triangle.
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Example of the Converse

• Determine whether a triangle with lengths 12, 20,
  and 16 form a right triangle.

This is a right triangle. A set of integers such
as 12, 16, and 20 is a Pythagorean triple.
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Converse Examples

• Determine whether
• 4, 5, 6 is a Pythagorean triple.
• Determine whether
• 15, 8, and 17 is a Pythagorean triple.

4, 5, and 6 is not a Pythagorean triple.
15, 8, and 17 is a Pythagorean triple.
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A Pythagorean Triple Is Any 3 Integers That Form
A Right Triangle
5, 12, 13 Multiples Family 10,24,26 25,60,65 35,8
4,91
3, 4, 5 Multiples Family 6,8,10 30,40,50 15,20,25
Multiples of Pythagorean Triples are also Pyth
Triples.
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Verifying Right Triangles

• The triangles on the right appear to be right
  triangles.
• Tell whether they are right triangles or not.

 
 
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Verifying Right Triangles

•  

 
?
?
The triangle is a right triangle.
Note squaring a square root!!
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Verifying Right Triangles

•  

 
?
?
?
The triangle is NOT a right triangle.
Note squaring an integer square root!!
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What Kind of Triangle??

• You can use the Converse of the Pythagorean
  Theorem to verify that a given triangle is a
  right triangle or obtuse or acute.

What Kind Of Triangle ? c² ?? a² b²
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Triangle Inequality
What Kind Of Triangle ? c² ?? a² b²
If the c² a² b² , then right If the c² gt a²
b² then obtuse If the c² lt a² b², then acute
The converse of the Pythagorean Theorem can be
used to categorize triangles.
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The converse of the Pythagorean Theorem can be
used to categorize triangles.
If a2 b2 c2, then triangle ABC is a right
triangle.
If a2 b2 lt c2, then triangle ABC is an obtuse
triangle.
If a2 b2 gt c2, then triangle ABC is an acute
triangle.
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Triangle Inequality

• 38, 77, 86
• c2 ? a2 b2
• 862 ? 382 772
• 7396 ? 1444 5959
• 7395 gt 7373

• Compare c2 with a2 b2
• Substitute values
• Square add
• c2 is greater than a2 b2
• The triangle is obtuse

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Triangle Inequality

• Compare c2 with a2 b2
• Substitute values
• Square add
• c2 is less than a2 b2
• The triangle is acute

• 10.5, 36.5, 37.5
• c2 ? a2 b2
• 37.52 ? 10.52 36.52
• 1406.25 ? 110.25 1332.25
• 1406.24 lt 1442.5

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4,7,9 9²__4² 7² 81__16 49 81 gt 65
OBTUSE
greater
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• 5,5,7
• 7² __5² 5²
• __ 25 25
• 49 lt 50
• ACUTE

SMALLER
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Building a foundation

• Construction You use four stakes and string to
  mark the foundation of a house. You want to make
  sure the foundation is rectangular.
• a. A friend measures the four sides to be 30
  feet, 30 feet, 72 feet, and 72 feet. He says
  these measurements prove that the foundation is
  rectangular. Is he correct?

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Building a foundation

• Solution Your friend is not correct. The
  foundation could be a nonrectangular
  parallelogram, as shown below.

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Building a foundation

• b. You measure one of the diagonals to be 78
  feet. Explain how you can use this measurement
  to tell whether the foundation will be
  rectangular.

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Building a foundation

• Because 302 722 782, you can conclude that
  both the triangles are right triangles. The
  foundation is a parallelogram with two right
  angles, which implies that it is rectangular

• Solution The diagonal divides the foundation
  into two triangles. Compare the square of the
  length of the longest side with the sum of the
  squares of the shorter sides of one of these
  triangles.