To verify that the quadrilateral ABCD is a parallelogram, we need to show that both pairs of opposite sides are parallel, which can be done by demonstrating that their corresponding vectors are parallel.
1. Calculate the vectors for sides AB, CD, AD, and BC:
- Vector AB: This vector goes from point A to point B. The coordinates of A are (-1, 1) and the coordinates of B are (3, 4). So, the vector AB is determined by subtracting the coordinates of A from B:
[tex]\[
AB = (B_x - A_x, B_y - A_y) = (3 - (-1), 4 - 1) = (4, 3)
\][/tex]
- Vector CD: This vector goes from point C to point D. The coordinates of C are (8, 4) and the coordinates of D are (4, 1). So, the vector CD is determined by subtracting the coordinates of C from D:
[tex]\[
CD = (D_x - C_x, D_y - C_y) = (4 - 8, 1 - 4) = (-4, -3)
\][/tex]
- Vector AD: This vector goes from point A to point D. The coordinates of A are (-1, 1) and the coordinates of D are (4, 1). So, the vector AD is determined by subtracting the coordinates of A from D:
[tex]\[
AD = (D_x - A_x, D_y - A_y) = (4 - (-1), 1 - 1) = (5, 0)
\][/tex]
- Vector BC: This vector goes from point B to point C. The coordinates of B are (3, 4) and the coordinates of C are (8, 4). So, the vector BC is determined by subtracting the coordinates of B from C:
[tex]\[
BC = (C_x - B_x, C_y - B_y) = (8 - 3, 4 - 4) = (5, 0)
\][/tex]
2. Check for parallelism of the opposite sides.
- Vectors AB and CD: Two vectors [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] are parallel if their cross product is zero, which is equivalent to saying:
[tex]\[
x_1 \cdot y_2 = y_1 \cdot x_2
\][/tex]
For vectors AB and CD:
[tex]\[
(4 \cdot -3) = (3 \cdot -4)
\][/tex]
[tex]\[
-12 = -12
\][/tex]
Therefore, AB is parallel to CD.
- Vectors AD and BC: Similarly, we check if vectors AD and BC are parallel:
[tex]\[
(5 \cdot 0) = (0 \cdot 5)
\][/tex]
[tex]\[
0 = 0
\][/tex]
Therefore, AD is parallel to BC.
Since both pairs of opposite sides are parallel, ABCD is indeed a parallelogram.
