Greenwood press

94
QUADRILATERALS
Using an inch ruler to divide
a board lengthwise into three
equal sections.

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A pantograph is used to produce a scale drawing. The picture below of a pan-
tograph shows that the four pivot points that connect the pieces of wood form a
parallelogram. The ratio of lengths of sides controls the scale of magnification.
Pantographs that handle three dimensions are used to trace solid models of bolts,
car fenders, or teeth. The pantograph records the three-dimensional coordinates
for the surface of the object. Milling machines use the database of coordinates to
shape a block of metal, plastic, or carbon composite into a high-precision copy
of the original object. 
The parallelogram law is used in physics to determine the net result of two
forces. The vectors 
a = (3, 3) and  b = (7,–1) are shown on the figure below as
arrows starting at the origin and ending at the respective coordinates. The paral-
lelogram law indicates that the resultant vector is found by completing the par-
allelogram defined by the vectors. The diagonal from the origin is the desired
vector. This corresponds to the point that would be found by the addition of coor-
dinates:
(3, 3) + (7,–1) = (10, 2). (See Vectors.)
Because of the many uses of quadrilaterals, students around the world are
expected to know formulas for the area and perimeter of most common quadri-
laterals. In addition, they must also know the volume formulas for the three-
dimensional analogs of some quadrilaterals such as the cube and rectangular
solid. Formulas for the multidimensional parallelepipeds are expressed as deter-
minants of matrices formed from the vectors. For example, the area of the paral-
lelogram illustrated above can be computed from the determinant of 

3
7
3 −1
= 3(−1) − 7(3) = −24. The area is 24. (See Matrices.) Extensions to more
dimensions provide measures of strength of association of variables in multi-
variate statistics.