Three consecutive vertices of a parallelogram ABCD are A (3, 0) ; B (5, 2) ,C (- 2, 6). Find the fourth vertex D.
CD will be parallel to AB, and of the same length.
B-A = (2,2)
So, D = C+(2,2) = (0,8)
or C-(2,2) = (-4,4)
To find the fourth vertex D of the parallelogram ABCD, we need to use the properties of a parallelogram, which states that opposite sides are parallel and equal in length.
Step 1: Find the vector connecting points A and B.
We can find this vector by subtracting the coordinates of point A from the coordinates of point B.
Vector AB = (5 - 3, 2 - 0) = (2, 2).
Step 2: Find the coordinates of point D.
Since the opposite sides of a parallelogram are parallel, the vector connecting points C and D should have the same direction as vector AB. Thus, vector CD should be equal to vector AB.
We can find the coordinates of point D using the coordinates of point C and vector AB.
Coordinates of point D = (x-coordinate of point C + x-component of vector AB, y-coordinate of point C + y-component of vector AB).
Coordinates of point D = (-2 + 2, 6 + 2) = (0, 8).
Therefore, the fourth vertex D of the parallelogram ABCD is D(0, 8).