Chords in a Coordinate System?

Question

A rectangular activities room floor is to be constructed in a sports complex. The floor has vertices A(-4,-11), B(12,21), C(36,9), and D(20,-23)

The area is to be sectioned off into squares (as roughly shown here: http://i17.tinypic.com/6gwzlac.jpg ) Detemrine the coordinates of the endpoints of each dotted dividing line using calculations and determine the coordinates of the center of the floor.

Answer 1

The midpoint of any line segment is just the average of the two endpoints' x and y values. The midpoint of (a,b) and (c,d) is ((a+c)/2,(b+d)/2).

The dashed line connecting sides AB to CD goes from the midpoint of AB to the midpoint of CD.

AB midpoint: ((-4+12)/2,((-11+21)/2)) =
(8/2,10/2) = (4,5)
CD midpoint: ((36+20)/2,(9+-23)/2) =
(56/2,-14/2) = (28,-7)
Endpoints: (4,5) and (28,-7)

The dashed line connecting sides BC to DA goes from the midpoint of BC to the midpoint of DA:

BC midpoint: ((12+36)/2,(21+9)/2) =
(48/2,30/2) = (24,15)
DA midpoint: ((20+-4)/2,(-23+-11)/2) =
(16/2, -34/2) = (8,-17)
Endpoints: (24,15) and (8,-17)

The center of the figure is the average of all four corners, or the midpoint of either dashed line.

Using the BC to DA line:
((24+8)/2, (15+-17)/2) = (32/2, -2/2) = (16,-1)

Using the AB to CD line:
((4+28)/2,(5+-7)/2)) = (32/2, -2/2) = (16,-1)

Using the four corners:
((-4+12+36+20)/4, (-11+21+9+-23)/4) =
(64/4, -4/4) = (16, -1)

Note that if the figure wasn't a perfect rectangle, you could still calculate the dashed lines the same way, but you'd have to calculate their intersection differently, because they wouldn't be guaranteed to meet at the center point of the figure.