Question

"Quadrilateral XMAS has vertices X(1,8), M(6,-1), and A(-4,-4). Determine and state the coordinates of S that would make XMAS a parallelogram. Prove that XMAS is a parallelogram."

a) S(-3,7) and show that opposite sides are parallel
b) S(8,1) and show that opposite sides are parallel
c) S(0,0) and show that opposite sides are parallel
d) S(5,4) and show that opposite sides are parallel

Answer 1

To make the quadrilateral XMAS a parallelogram, the coordinates of S should be (1, -13). To prove that XMAS is a parallelogram, we can show that opposite sides are parallel. The vectors representing the sides of the quadrilateral show that opposite sides AM and XS, and AS and XM, are parallel.

Step-by-step explanation:

To determine the coordinates of S that would make XMAS a parallelogram, we need to find the vector from X to M, and then add it to the coordinates of A. The vector from X to M can be found by subtracting the coordinates of X from the coordinates of M:
XM = M - X = (6, -1) - (1, 8) = (5, -9)
Then, we add this vector to the coordinates of A to find S:
S = A + XM = (-4, -4) + (5, -9) = (1, -13),

To prove that XMAS is a parallelogram, we need to show that opposite sides are parallel.
First, we find the vectors representing the sides of the quadrilateral:
AM = M - A = (6, -1) - (-4, -4) = (10, 3)
AS = S - A = (1, -13) - (-4, -4) = (5, -9)
XS = S - X = (1, -13) - (1, 8) = (0, -21)
XM = M - X = (6, -1) - (1, 8) = (5, -9)

We can see that AM and XS have the same x-component (10 and 0, respectively) and AS and XM have the same y-component (-9 and -9, respectively). Therefore, opposite sides AM and XS, and AS and XM, are parallel. This proves that XMAS is a parallelogram.