Pythagorean theorem

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Pythagorean theorem

• Section 5.3

• Pythagoras of Samos
• Lived about 569 BC - about 475 BC
• Greek philosopher
• Important ideas in
• Mathematics
• Astronomy
• Theory of music

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Definitions

• Hypotenuse- the side opposite the right angle
  (which is the longest side)
• Legs of a right triangle - the other two sides

Hypotenuse
Leg
Leg
3
(No Transcript)
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Theorem 5.4

• In a right triangle, the sum of the squares of
  the lengths of the legs is equal to the square of
  the length of the hypotenuse.
  a2 b2 c2

c
a
b
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THEOREM 5.4 Pythagorean Theorem
In a right triangle, the square of the length of
the hypotenuse is equal to the sum of the squares
of the lengths of the legs.
c 2 a 2 b 2
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Thus the sum of the squares on the smaller two
sides equals the square on the biggest side.
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Example 1

• Find the value of x. Leave the answer in simplest
  radical form.
• a2 b2 c2
• 82 x2 202
• 64 x2 400
• x2 336
• x

20
8
x
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What type of triangle is it?

• Statement Type
• c2 a2 b2 Right Triangle
• c2 lt a2 b2 Acute Triangle
• c2 gt a2 b2 Obtuse Triangle
• Example 1 Example 2
  Example 3
• 3, 4, 5 3, 4, 6
  5, 10, 11
• c2 ? a2 b2 c2 ? a2 b2 c2 ?
  a2 b252 ? 32 42 62 ? 32 42 112 ?
  52 102
• 25 ? 9 16 ______________
  ______________
• 25 25 ______________
  ______________Right Triangle
  _______ Triangle ________ Triangle

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Theorem 5.5

• Converse of the Pythagorean Theorem
• If the square of the length of one side is equal
  to the squares of the lengths of the other two
  sides then the triangle is a right triangle.
• Lengths 5, 12 and 13

?
13
12
?
5
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Pythagorean Triples
A Pythagorean triple consists of three positive
integers a, b, and c, such that a2  b2  c2.
(Memorize the first row. Page 258)
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Short-cuts are Useful!

• Method 1 Method 2
• a2 b2 c2

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6
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Example 2

• Brian is out hunting deer. He sees a large buck
  across the lake. He knows from the deer to the
  big pine is 75 and he is 25 from the big pine.
  How far is he from the deer assuming from the
  deer to the pine to Brian is 90 degrees?

75 feet
25 feet
? feet
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Solution

• a2 b2 c2
• 252 752 c2
• 625 5625 c2
• 6250 c2 (Take the square
  root of each side.)
• 79 c
  (Rounded to nearest whole number.)
• Conclusion ?
• Brian is about 79 feet from the deer.

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Homework

• Day 1 Page 259 1 22
• Day 2 Page 260 25-30,33-36,56,58