What is the area of the parallelogram ABCD with vertices at (2, 2), (6,10), (8, 5) and (12, 13)

To find the area of a parallelogram given its vertices, we can use the formula based on the cross product of two vectors formed by its adjacent sides. The vertices provided are \( A(2, 2) \), \( B(6, 10) \), \( C(8, 5) \), and \( D(12, 13) \).

The steps are as follows:

1. **Form two vectors** representing two adjacent sides of the parallelogram:
    - Vector \( \overrightarrow{AB} = B - A = (6 - 2, 10 - 2) = (4, 8) \)
    - Vector \( \overrightarrow{AC} = C - A = (8 - 2, 5 - 2) = (6, 3) \)

2. **Compute the cross product** of these vectors in two dimensions. The magnitude of the cross product gives the area of the parallelogram:
   \[
   \text{Area} = | \overrightarrow{AB} \times \overrightarrow{AC} |
   \]
   In 2D, the cross product is calculated as:
   \[
   \overrightarrow{AB} \times \overrightarrow{AC} = |(4 \times 3) - (8 \times 6)| = |12 - 48| = |-36| = 36
   \]

Thus, the area of the parallelogram is **36 square units**.

Would you like more details, or have any further questions?

Here are some related questions:
1. How do you find the cross product in 3D?
2. What is the difference between a parallelogram and a rectangle?
3. How can you find the perimeter of a parallelogram using vertex coordinates?
4. What is the formula for the area of a triangle using its vertices?
5. How do vector operations relate to geometric properties like area?

**Tip:** The cross product is a great way to find areas of parallelograms and triangles in vector spaces!

What is the area of the parallelogram ABCD with vertices at (2, 2), (6,10), (8, 5) and (12, 13)?

Learn how to calculate the area of a parallelogram with vertices at (2, 2), (6, 10), (8, 5), and (12, 13) using vector cross product. This step-by-step solution explains the geometric and vector operations involved. Suitable for high school students in Grades 9-12.

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