A Question Of Quadrilaterals

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Term Paper Title	A Question Of Quadrilaterals
# of Words	1033
# of Pages (250 words per page double spaced)	4.13

A Question of Quadrilaterals



     My colleagues and I are beginning a new business.  I have recently bought a lot on
where my business will be located.  In the next few days I need to show my colleagues a
diagram of the lot.  The original map that I received is too small.  In order to construct a
larger map, I must first construct the original points on a graph.  The coordinates of the
original map of the lot are:  (1,4), (5,0), (5,8), and (9,4).  My shape of the land is shown
below:



     I also need to classify the shape of my lot.  There are two aspects you need to look
at when classifying:  the distance of each line, and the slope of each line.  Judging by
appearance, my lot appears to be in the shape of a square.  One way to be exact is to
measure the lengths of the lines and make sure that they are congruent using the distance
formula.
     AB=        d=  (x2-x1)2 + (y2-y1)2          BC=        d=  (9-5)2 + (4-0)2
       (1,4) (5,0)   d=  (5-1)2 + (0-4)2                      (5,0) (9,4)   d=  16+16
             d=  16+16                            d=  32  = 4  2
             d=  32 = 4  2

     CD=        d=  (9-5)2 + (4-0)2                DA=        d=  (1-5)2 + (4-8)2
      (9,4) (5,8)    d=  16+16                     (5,8) (1,4)       d=  16+16
             d=  32 = 4  2                               d=  32 = 4  2

     Now that I have determined that all of the sides are congruent, d= 4  2, I must find
that the slopes are the same.  To be a square, they must equal 90 .
               slope formula:  m= y2-y1
                                     x2-x1
     AB=        m= 0-4 = -4 =                BC=        m= 4-0 = 4 =
      (1,4) (5,0)         5-1     4        -1                  (5,0) (9,4)              9-5    4       1

     CD=        m= 8-4 = 4  =                              DA=        m= 4-8 = -4 =
      (9,4) (5,8)         5-9    -4         -1                (5,8) (1,4)         1-5     -4          1

     Using the slope formula I have determined that the lot is square.  The reason is
because the slopes which equal 1 are parallel, and -1 are parallel.  Therefore, the slopes of
1, and -1 would be perpendicular forming 90  angles.  I decided once the lot was graphed,
it was too hard to view clearly.  I performed a dilation (enlargement) for the map, and
enlarged it by two.  I used matrices to multiply them.  By putting the points in a matrix
and multiplying them by two, made the coordinates for the dilation.

          A  B  C  D               A’  B’  C’  D’
x coordinate     1   5   9   5  x2 =     x    2   10  18     10       A’ (2,8)     B’ (10,0)
y coordinate     4   0   4   8          y    8    0    8    16       C’ (18,8)     D’ (10,16)



     The new figure, I feel, is the same type ...

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