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
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.
To find the coordinates of a point D such that ABCD forms a parallelogram, we can use the properties of parallelograms.
(a) If AB is one of the diagonals:
1. Find the midpoint of AB. The midpoint formula is:
Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)
Let's find the midpoint of AB:
Midpoint of AB = ((4 + 5) / 2, (3 + 8) / 2) = (4.5, 5.5)
2. Once you have the midpoint, you can use it to find the coordinates of D. Since ABCD is a parallelogram, the diagonal BD is parallel to AC. Therefore, the coordinates of D can be found by adding the x-coordinate and y-coordinate differences of A and C to the x-coordinate and y-coordinate of the midpoint of AB.
x-coordinate of D = x-coordinate of midpoint of AB + (x-coordinate of C - x-coordinate of A)
y-coordinate of D = y-coordinate of midpoint of AB + (y-coordinate of C - y-coordinate of A)
Let's calculate the coordinates of D:
x-coordinate of D = 4.5 + (3 - 4) = 4.5 - 1 = 3.5
y-coordinate of D = 5.5 + (10 - 3) = 5.5 + 7 = 12.5
Therefore, the coordinates of D are (3.5, 12.5).
(b) If AC is one of the diagonals:
1. Follow the same steps as in part (a) to find the midpoint of AC. Let's calculate it:
Midpoint of AC = ((4 + 3) / 2, (3 + 10) / 2) = (3.5, 6.5)
2. Use the midpoint of AC to find the coordinates of D. Since ABCD is a parallelogram, the diagonal BD is parallel to AB. Therefore, the coordinates of D can be found by adding the x-coordinate and y-coordinate differences of B and A to the x-coordinate and y-coordinate of the midpoint of AC.
x-coordinate of D = x-coordinate of midpoint of AC + (x-coordinate of B - x-coordinate of A)
y-coordinate of D = y-coordinate of midpoint of AC + (y-coordinate of B - y-coordinate of A)
Let's calculate the coordinates of D:
x-coordinate of D = 3.5 + (5 - 4) = 3.5 + 1 = 4.5
y-coordinate of D = 6.5 + (8 - 3) = 6.5 + 5 = 11.5
Therefore, the coordinates of D are (4.5, 11.5).
(c) If BC is one of the diagonals:
1. Follow the same steps as in part (a) to find the midpoint of BC. Let's calculate it:
Midpoint of BC = ((5 + 3) / 2, (8 + 10) / 2) = (4, 9)
2. Use the midpoint of BC to find the coordinates of D. Since ABCD is a parallelogram, the diagonal AD is parallel to BC. Therefore, the coordinates of D can be found by adding the x-coordinate and y-coordinate differences of A and B to the x-coordinate and y-coordinate of the midpoint of BC.
x-coordinate of D = x-coordinate of midpoint of BC + (x-coordinate of A - x-coordinate of B)
y-coordinate of D = y-coordinate of midpoint of BC + (y-coordinate of A - y-coordinate of B)
Let's calculate the coordinates of D:
x-coordinate of D = 4 + (4 - 5) = 4 - 1 = 3
y-coordinate of D = 9 + (3 - 8) = 9 - 5 = 4
Therefore, the coordinates of D are (3, 4).
In summary:
(a) If AB is one of the diagonals, D(3.5, 12.5)
(b) If AC is one of the diagonals, D(4.5, 11.5)
(c) If BC is one of the diagonals, D(3, 4)