Question: User: T: Quadrant 1
9) Vertices: A(3, 3), B(5, 7), C(7, 7), D(8, 3)
4 units
5 + 2
2 \cdot 4 = 7
14 units^2
5 units
10) Vertices: A(3, 5), B(2, 8), C(9, 8), D(8, 5), E
(4, 7)
7 units
3 units
18
1/2 \cdot 5 \cdot 2 = 5
18 - 5 = 13 units^2
11) Vertices: A(0, 0), B(-3, 2), C(0, 4), D(3, 2),
d1 = 6
d2 = 4
1/2 \cdot 6 \cdot 4 = 12
1/2 \cdot 2 \cdot 3 = 3
12 - 3 = 9
9 square units
12) Vertices: A(-1, 4), B(2, 3), C(2, -4), D
(-1, -3), E(-1, -2), F(-4, -3), G(-4, 2), H
(-1, 3)
6 units
4 units
2 units
3 units
4 units
3 units
3 units
2 units
Can you give me the answer to the last answer with the work shown of how you got the answer
1. **Problem Statement:** Find the area of the polygon with vertices A(-1, 4), B(2, 3), C(2, -4), D(-1, -3), E(-1, -2), F(-4, -3), G(-4, 2), H(-1, 3).
2. **Method:** Use the Shoelace formula (also called Gauss's area formula) for polygons.
3. **Shoelace formula:** For vertices $(x_1,y_1), (x_2,y_2), ..., (x_n,y_n)$ in order, area = $$\frac{1}{2} \left| \sum_{i=1}^{n} (x_i y_{i+1} - y_i x_{i+1}) \right|$$ where $(x_{n+1}, y_{n+1}) = (x_1, y_1)$.
4. **List vertices in order:**
$A(-1,4), B(2,3), C(2,-4), D(-1,-3), E(-1,-2), F(-4,-3), G(-4,2), H(-1,3)$
5. **Calculate the sums:**
Calculate $\sum x_i y_{i+1}$:
$(-1) \times 3 = -3$
$2 \times (-4) = -8$
$2 \times (-3) = -6$
$(-1) \times (-2) = 2$
$(-1) \times (-3) = 3$
$(-4) \times 2 = -8$
$(-4) \times 3 = -12$
$(-1) \times 4 = -4$
Sum: $-3 -8 -6 + 2 + 3 -8 -12 -4 = -36$
Calculate $\sum y_i x_{i+1}$:
$4 \times 2 = 8$
$3 \times 2 = 6$
$-4 \times (-1) = 4$
$-3 \times (-1) = 3$
$-2 \times (-4) = 8$
$-3 \times (-4) = 12$
$2 \times (-1) = -2$
$3 \times (-1) = -3$
Sum: $8 + 6 + 4 + 3 + 8 + 12 - 2 - 3 = 36$
6. **Apply formula:**
$$\text{Area} = \frac{1}{2} |(-36) - 36| = \frac{1}{2} |-72| = 36$$
7. **Answer:** The area of the polygon is $36$ square units.