To find the area of the region covered by Group A, we will use the coordinates provided: (1, 1), (-3, 2), (-2, -2), and (1, -3). We can divide the area into two triangles and calculate the area of each triangle separately.
### Step 1: Identify the coordinates
The coordinates of the vertices of the polygon are:
- A(1, 1)
- B(-3, 2)
- C(-2, -2)
- D(1, -3)
### Step 2: Divide the polygon into triangles
We can divide the quadrilateral into two triangles:
1. Triangle 1: A(1, 1), B(-3, 2), C(-2, -2)
2. Triangle 2: A(1, 1), C(-2, -2), D(1, -3)
### Step 3: Calculate the area of Triangle 1 (A, B, C)
Using the formula for the area of a triangle given by vertices (x1, y1), (x2, y2), (x3, y3):
\[
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\]
Substituting the coordinates of Triangle 1:
- \(x_1 = 1, y_1 = 1\)
- \(x_2 = -3, y_2 = 2\)
- \(x_3 = -2, y_3 = -2\)
\[
\text{Area}_1 = \frac{1}{2} \left| 1(2 - (-2)) + (-3)(-2 - 1) + (-2)(1 - 2) \right|
\]
\[
= \frac{1}{2} \left| 1(4) + (-3)(-3) + (-2)(-1) \right|
\]
\[
= \frac{1}{2} \left| 4 + 9 + 2 \right| = \frac{1}{2} \left| 15 \right| = \frac{15}{2} = 7.5 \text{ square units}
\]
### Step 4: Calculate the area of Triangle 2 (A, C, D)
Using the same formula for Triangle 2:
- \(x_1 = 1, y_1 = 1\)
- \(x_2 = -2, y_2 = -2\)
- \(x_3 = 1, y_3 = -3\)
\[
\text{Area}_2 = \frac{1}{2} \left| 1(-2 - (-3)) + (-2)(-3 - 1) + 1(1 - (-2)) \right|
\]
\[
= \frac{1}{2} \left| 1(1) + (-2)(-4) + 1(3) \right|
\]
\[
= \frac{1}{2} \left| 1 + 8 + 3 \right| = \frac{1}{2} \left| 12 \right| = \frac{12}{2} = 6 \text{ square units}
\]
### Step 5: Calculate the total area
Now, we add the areas of both triangles:
\[
\text{Total Area} = \text{Area}_1 + \text{Area}_2 = 7.5 + 6 = 13.5 \text{ square units}
\]
### Final Answer
The area of the region covered by Group A is **13.5 square units**.
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