The coordinates of three of the vertices of parallelogram ABCD are A(2, 0), B(4, 3), C(3, 1). How many possibilities are there for the position of the fourth vertex? What are the coordinates of the fourth vertex?
Answers
Answer:
The three of the vertices of the parallelogram ABCD are given as
A(x1, y1) = (2, 0)
B(x2, y2) = (4, 3)
C(x3, y3) = (3, 1)
Let the coordinates of the fourth vertex D be (x4, y4).
Case 1: Finding the possible positions of the fourth vertex of parallelogram ABCD:
A parallelogram is a quadrilateral, with two pairs of parallel sides and Opposite angles equal and the opposite or facing sides of a parallelogram are equal in length.
Depending on the definition of the parallelogram above and the given coordinates in the question, we can draw three possible parallelograms by joining the vertices as shown in the figure.
Thus, there are 3 possible positions (D, D’, D’’) for the fourth vertex of the parallelogram ABCD.
This can be clearly understood from the figure attached below.
Case 2: Finding the coordinates of one of the fourth vertex of parallelogram ABCD:
The midpoint of a line segment joining two points (x1,y1) & (x2,y2) = [(x1+x2)/2, (y1+y2)/2]
∴ The midpoint of AC
= [(x1+x3)/2, (y1+y3)/2]
= [(2+3)/2, (0+1)/2]
= (5/2, 1/2) …… (i)
We know that the diagonals of a parallelogram bisect each other and each diagonal separates into two congruent triangles.
∴ Midpoints of BD = Midpoints of AC
⇒ [(x2+x4)/2, (y1+y4)/2] = (5/2, 1/2) ….. [from (i)]
⇒ [(4+x4)/2, (3+y4)/2] = (5/2, 1/2)
⇒ [(4+x4)/2] = 5/2 or [(3+y4)/2] = ½
⇒ x4 = 5-4 = 1 or y4 = 1-3 = -2
Hence, the coordinates of vertex D(x4, y4) = (1, -2).
