Areas and the 
Pythagorean Theorem


Assignment #4.  (posted November 21st)

  1.  Area Puzzles.  Cut out the figures and reassemble them to form rectangles.  Use the rectangles to find the area formula of the figure.

  2.  

    Parallelogram

    b = base;  h = height
    Kite

    w = width;  h = height
    Regular Pentagon

    p = perimeter 
    r = distance from the center to the middle of a side
    Regular Hexagon


    p = perimeter 
    r = distance from the center to the middle of a side

    Regular Heptagon


    p = perimeter 
    r = distance from the center to the middle of a side

    A regular polygon is one in which every side and every angle is the same.  The square, regular pentagon, hexagon, and heptagon are examples.  Based on your last three answers, explain why the area of any regular polygon is

    rp / 2

    Does this agree with the fomula for the area of a circle?
    How about a square?


     
  3. Chinese proof of the Pythagorean Theorem.  Make a proof of the Pythagorean Theorem using this right triangle.  You'll need to make a square whose side matches the hypotenuse of the triangle, then use the Chinese decomposition to cut it up into four triangles and a square, and finally re-form the pieces into a square of side next to a square of side b (the ``Utah'' shape).


  4. Thabit's proof of the Pythagorean Theorem.  Make a proof of the Pythagorean Theorem for the following triangle using Thabit's method.  You'll need to start by making two squares, one of side a, and one of side b.


  5.  
  6. Pythagorean triples.  Find the Pythagorean Triples generated by the following values of x and y.
    1. x = 2   and   y = 1
    2. x = 4   and   y = 3
    3. x = (your age)   and   y =  (the number of the month in which you were born)
  7. Primitive Pythagorean triples.
    1. Which of the triples you found in the previous problem are primitive?
    2. For which values of  x is the triple generated by    x = (a whole number n)   and   y =  1  primitive?
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Rachel W. Hall / Department of Math and Computer Science / St. Joseph's University / rhall@sju.edu