Find the area of this quadrilateral with the given vertices:
A(-8,6) B(-5,8) C(-2,6) D(-5,0)
To find the area of a quadrilateral with the given vertices, you can calculate the area using the shoelace formula. The shoelace formula, also known as Gauss's area formula or the surveyor's formula, calculates the area of a polygon given the coordinates of its vertices.
The formula requires arranging the coordinates of the vertices in a specific order and then performing some arithmetic operations. Let's calculate the area step by step:
1. Arrange the coordinates of the vertices A, B, C, and D in a counterclockwise direction:
A(-8,6), B(-5,8), C(-2,6), D(-5,0)
By keeping the vertices in counterclockwise order, we ensure that the quadrilateral forms a closed shape.
2. Write down the coordinates in a matrix:
| x | y |
|----|----|
| -8 | 6 |
| -5 | 8 |
| -2 | 6 |
| -5 | 0 |
3. Multiply the values in each column diagonally and sum them:
(-8 * 8) + (-5 * 6) + (-2 * 0) + (-5 * 6) = -64 - 30 + 0 - 30 = -124
4. Multiply the values in each column diagonally in the opposite direction and sum them:
(6 * -5) + (8 * -2) + (6 * -5) + (0 * -8) = -30 - 16 - 30 + 0 = -76
5. Calculate the absolute value of the difference between the two sums:
|(-124) - (-76)| = |-124 + 76| = |-48| = 48
6. Divide the absolute value by 2:
48 / 2 = 24
The area of the quadrilateral with vertices A(-8,6), B(-5,8), C(-2,6), D(-5,0) is 24 square units.