MATH 251-003 EXTRA-CREDIT ASSIGNMENT:
ON GEOMETIC CENTERS AND CENTROIDS
1. Use partitioning and the fact that the centroid of a triangular region is located
2
rds
3
of
the way from a vertex to the midpoint of its opposite side to prove that the centroid of
the trapezoid:
(d,f)
(c,f)
(a,e)
is at
(b,e)
C = ( x ,y )
where
(b 2 + bc + c 2 ) − (a 2 + ad + d 2 )
3 [(b + c) − (a + d )]
x =
Then upon setting
trapezoid:
b=c
in
(1)
, y =
x =
(c,f)
(c,e)
(a,e)
C = ( x, y )
(1)
it follows immediately that the centroid of the
(d,f)
is at
(c − d )(e + 2 f ) + (b − a)(2e + f )
3 [(b + c) − (a + d )]
where
3c 2 − (a 2 + ad + d 2 )
3 [ 2c − ( a + d ) ]
, y =
(c − d )(e + 2 f ) + (b − a)(2e + f )
3 [ 2c − (a + d ) ]
(2)
By symmetry it must also be the case that the centroids of the trapezoids:
(d,f)
(c,f)
(a,e)
(b,e)
(d,f)
(c,f)
(a,e)
are located at
C = ( x, y )
where
(b,e)
x and y are still given by ( 1 ). Thus, in every case:
x = F ( a, b, c, d ) =
(b 2 + bc + c 2 ) − (a 2 + ad + d 2 )
3 [(b + c) − (a + d )]
and
y = G ( a, b, c, d, e, f ) =
(c − d )(e + 2 f ) + (b − a)(2e + f )
3 [(b + c) − (a + d )]
2. Now to approximate the geometric center or centroid of the
state of Illinois, proceed as follows:
( i ) Estimate the state’s area with 1 triangular and 18
trapazoidal subregions. Their vertices should be
determined by the map displayed to the right. Take the
( x , y ) coordinates of the lower tip of the state to be at
( – 85 , 0 ), and take the northeastern vertex of the state
to be located at ( – 13 , 380 ).
All distances are approximate and are in miles. The given
east – west distances are 20 miles apart. You will also
need the distances between the eastern boundary of the
state and the line running north/south that forms the
eastern boundary of the center of the state. Beginning
with the northernmost dimensions, the distances are 13
and 10 miles, and beginning with the southernmost
dimensions, the distances are 85 (southern tip), 50, 30,
25, 15, and 10 miles. Assume that all other east/west
dimensions are measured from the easternmost boundary.
( ii ) Find the centroid of each of the 19 polygonal subregions determined in ( i )
by utilizing the formulas derived in problem 1.
( iii ) Form the appropriate convex combination of the centroids of the polygonal
subregions obtained in ( ii ) to determine the geometric center of the state.
( iv ) On the Illinois County map shown below, determine the county which includes
the center of the state based on your coordinates in (iii).