Three vertices of a parallelogram WXYZ are X(-2, -3) Y(0, 5) and Z(7, 7). Find the coordinates of vertex W.
a
(4, 0)
b
(5, 0)
c
(9, 15)
d
(5, -1)
WZ must be equal to and parallel to XY
Y-X = (2,8)
so, Z-W = (2,8)
W = Z-(2,8) = (5,-1)
To find the coordinates of vertex W, we need to determine the fourth vertex of the parallelogram.
Given that opposite sides of a parallelogram are parallel, we can calculate the vector representing the displacement between points X and Y. Since W and Z are also connected by a parallel side, the same displacement vector can be used to find the coordinates of vertex W.
The displacement vector can be found by subtracting the coordinates of point X from the coordinates of point Y:
Displacement vector = Y - X
= (0, 5) - (-2, -3)
= (0 - (-2), 5 - (-3))
= (2, 8)
To find the coordinates of vertex W, we add the displacement vector to the coordinates of point Z:
W = Z + displacement vector
= (7, 7) + (2, 8)
= (7 + 2, 7 + 8)
= (9, 15)
Therefore, the coordinates of vertex W are (9, 15).
The correct answer is option c) (9, 15).
To find the coordinates of vertex W, we need to use the fact that opposite sides of a parallelogram are equal in length and parallel to each other.
Given that Y is the midpoint of WX (since it is the midpoint of the diagonal), we can find the coordinates of W by using the formula for finding the midpoint between two points. The midpoint formula is:
Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)
Let's calculate the coordinates of W using point X(-2, -3) and Y(0, 5):
Midpoint of WX = ((-2 + 0) / 2, (-3 + 5) / 2)
= (-1 / 2, 2 / 2)
= (-1/2, 1)
Therefore, the coordinates of vertex W are (-1/2, 1).
None of the given options matches these coordinates, so the correct answer is not listed.