Question

Let A(4,3) B(5,8) and C(3,10) be three points in a coordinate plane. Find the coordinates of a point D such that the points ABC and D form a parallelogram with (a)AB as one of the diagonals
(b)AC as one of the diagonals
(c)BC as one of the diagonals

Answer 1

Plot the three points on graph paper or on your calculator.

To insert point D as the 4th point of the parallelogram with AC as diagonal means that you need to find lines L1 parallel to AB passing through C, and L2 parallel to BC passing through A. The intersection of L1 and L2 will be the fourth vertex of the parallelogram.

To find the line L1 parallel to AB passing through C:
Calculate slope of line AB:
slope AB = (yb-ya)/(xb-xa)
=(8-3)/(5-4)=5
L1 passes through C(3,10), so
L1 : y-10 = 5(x-3)
or
L1: y = 5x - 5
Similarly find L2:
slope of BC = (10-8)/(3-5)=2/(-2)=-1
L2 passes through A(4,3), so
L2 : y-3 = -1(x-4), or
L2 : y = -x + 7

The 4th vertex is the intersection of L1 and L2, solved by the system of equations L1 and L2:
y=5x-5...(1)
y=-x+7...(2)
subtract:
6x-12=0, therefore x=2, y=5, or the point D is (2,5).

You can find the fourth vertex of the parallelogram with BC as diagonal in a similar way.