Final answer:
The area of a parallelogram with vertices A(1,2), B(4,3), C(3,7), and D(6, 8) is calculated using the determinant of the cross product of vectors AB and AD, resulting in an area of 13 square units.
Step-by-step explanation:
To find the area of a parallelogram given its vertices A(1,2), B(4,3), C(3,7), and D(6, 8), one method is to use the formula involving the cross product of vectors. You can treat two adjacent sides of the parallelogram as vectors and then calculate the cross product of these two vectors to find the area. The area of the parallelogram is then the magnitude of this cross product.
We can start by finding the vectors AB and AD:
- Vector AB = (Bx - Ax, By - Ay) = (4 - 1, 3 - 2) = (3, 1)
- Vector AD = (Dx - Ax, Dy - Ay) = (6 - 1, 8 - 2) = (5, 6)
Next, we find the cross product of AB and AD which gives us the area of the parallelogram:
Area of parallelogram = |AB x AD| = |(3, 1) x (5, 6)|
To calculate the cross product, we will use the determinant of a 2x2 matrix that includes the components of vectors AB and AD.
The determinant |AB x AD| is calculated as:
|AB x AD| = (3 * 6) - (5 * 1) = 18 - 5 = 13
The area of the parallelogram is the absolute value of this determinant which means the area is 13 square units.