Find the area of quadrilateral whose verties are A(0,2), B(2,7), C(6,10), and D (9,-2).
There is an old method, called Pick's theorem: https://en.wikipedia.org/wiki/Pick%27s_theorem
So graph the points on graph paper, count the points, and use the formula.
(AB)^2 = (2-0)^2 + (7-2)^2 = 29,
AB = 5.39.
(BC)^2 = (6-2)^2 + (10-7)^2 = 25,
BC = 5.
(CD)^2 = (9-6)^2 + -2-6)^2 = 73,
CD = 8.54.
(AD)^2 = ((9-0)^2 + (-2-2)^2 =
AD =
P = AB + BC + CD + AD =
A^2 = (P*AB*BC*CD*AD)/16 =
A =
To find the area of a quadrilateral with given coordinates, you can use the Shoelace Formula (also known as the Gauss-Green Theorem or the surveyor's formula).
1. Begin by listing the coordinates of the vertices in clockwise order. In this case, the given vertices are A(0,2), B(2,7), C(6,10), and D(9,-2).
2. Write down the coordinates in a column, repeating the first coordinate at the bottom.
```
A(0,2)
B(2,7)
C(6,10)
D(9,-2)
A(0,2)
```
3. Multiply diagonally and add the products:
```
(0 * 7) + (2 * 6) + (2 * 9) + (2 * 10) + (2 * 9)
```
4. Multiply diagonally in the opposite direction and subtract those products:
```
(2 * 2) + (7 * 6) + (6 * 9) + (10 * 2) + (9 * 0)
```
5. Take the absolute value of the result:
```
(0 * 7) + (2 * 6) + (2 * 9) + (2 * 10) + (2 * 9) - [(2 * 2) + (7 * 6) + (6 * 9) + (10 * 2) + (9 * 0)]
```
6. Simplify the expression:
```
0 + 12 + 18 + 20 + 18 - 4 - 42 - 54 - 20 + 0
```
```
-12
```
7. Take the absolute value to get the area:
```
|-12| = 12
```
Therefore, the area of the quadrilateral is 12 square units.