Pythagorean Theorem

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Pythagorean Theorem

• By Amy Ehlers
• aehlers_at_nmu.edu

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Basic Concepts Angles

• Definition of an angle two lines that intersect
  at a vertex, and the span between the two lines
  is called the angle.
• This top figure shows the angle N and the two
  lines A and B.

• This bottom figure shows a perfect 90 degree
  angle, or a right angle with lines A and B.

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Basic Concepts Triangle

• Definition of a triangle any geometric shape
  consisting of three points or vertices which are
  connected my straight line segments called sides.
  
• When one of the angles in the triangle is a
  right angle, than the triangle is called a right
  triangle.
• There are three sides to every triangle.
• Side a and b are called the legs, and side
  c is called the hypotenuse.
• Sides a and b can often be switched around,
  but side c must always be the longest side, and
  the side opposite the right angle.

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What is the Pythagorean Theorem?

• The Pythagorean Theorem is a statement that only
  works on right triangles.
• The theorem states that The area of the
  square built upon the hypotenuse of a right
  triangle is equal to the sum of the areas of the
  squares upon the remaining sides.
• Sounds confusing right? So lets break this
  statement down so it is easier to understand.

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Understanding The Pythagorean Theorem

• According to the theorem, the sum of the area of
  the two pink squares, A and B, is equal to
  the area of the larger blue square, C.

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Basic Concept Area of a Square

• In the theorem, two smaller squares are built
  off of each leg, and one larger square is built
  of the hypotenuse.
• Finding the area of a square is easy, because
  all of the sides on a square are equal or
  congruent.
• As long as you know one of the sides of a
  square, it is easy to find the area.
• To calculate the area of a square, take the
  measure of one of the sides, (in the square to
  the left it would be 4) and square it. The area
  of the square pictured is 42 which equals 4X4 or
  16.

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Understanding The Pythagorean Theorem

• To show how the Pythagorean theorem works, lets
  take a look at the figure to the left.
• On square a, the length of the side of the
  square, which is also one of the legs of the
  triangle is 3. To find the area of that square
  you have to square 3, to get 9.
• On square b, the length of the side of the
  square, which is also one of the legs of the
  triangle is 4. To find the area of that square
  you must square 4 to get 16.
• The area of a plus the area of b equals 25.
• On square c, the length of the side of the
  square, which is also the length of the
  hypotenuse, is 5. To find the area of the square
  you have to square 5 to get 25.
• Therefore the Pythagorean theorem must be true,
  because the sum of the area of the two squares
  built off the legs equals the area of the square
  built of the hypotenuse.

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Understanding The Pythagorean Theorem

• Now if you are still having problems
  understanding the Pythagorean theorem, take a
  look at the website below.
• Pythagorean Puzzle
• This link takes you to an interactive puzzle,
  that helps you to understand that the sum of the
  area of the squares built off the legs of the
  triangle, is equal to the area of the square
  built off the hypotenuse.

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The Pythagorean Theorem

• Now that you understand the concept behind the
  Pythagorean Theorem, lets take more of an
  algebraic look at it.
• As previously stated, the sum of the area of the
  two pink squares, A and B, is equal to the
  area of the larger blue square, C.
• Area of square A a2
• Area of square B b2
• Area of square C c2
• Thus algebraically the Pythagorean theorem
  states
• a2 b2 c2
• In a right triangle, a and b are the legs,
  and c is the hypotenuse.

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Will I ever use this in real life?

• Whenever we learn a new concept, the question
  that is always asked, is will we ever use this in
  real life?
• The answer. Of course you will
• Take a look at some of the following examples
  that will show how you can use the Pythagorean
  theorem in real life.

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Using The Pythagorean Theorem

• It is Timmys dream to get his kite to reach the
  roof of his house.
• Timmy knows that his house is 12 feet high, and
  that he will be standing 5 feet from the house.
• What Timmy doesnt know is how much string he
  will need for his dream to come true.
• How will Timmy solve this problem?
• If you take a closer look, you will notice that
  the side of the house, and the distance that
  Timmy is standing from the house, form the legs
  of a right triangle.
• To find out how much string Timmy will need so
  his kite can touch the roof, you will need to use
  the Pythagorean Theorem.

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Using The Pythagorean Theorem

• First step in using the Pythagorean theorem is
  to plug the numbers into the equation. Then work
  out the equation.
• a2 b2 c2
• 122 52 c2
• 144 25 c2
• 169 c2
• c
• 13 c
• Therefore Timmy needs 13 feet of string so that
  his kite will reach the top of the house.

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Using The Pythagorean Theorem

• You can also use Pythagoras to find out
  distances on a map.
• Looking at the map you are at the x, the pizza
  shop is 3 miles from where you are.
• you also know that from were you are standing
  now, your friends house is 4 miles.
• You need to go to the pizza shop first to pick
  up a pizza, and then you want to take a direct
  path to your friends house, but you are not sure
  how long that will take and you need to let your
  friend know what time you will be showing up at
  her house.

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Using The Pythagorean Theorem

• Use the Pythagorean theorem to solve this
  problem.
• a2 b2 c2
• 32 42 c2
• 9 16 c2
• 25 c2
• c
• 5 c
• Therefore the direct path from the pizza shop to
  the friends house will take 5 miles.