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The problem asks to find the perimeter and area of quadrilateral \( ABCD \), using the coordinates given on the grid.

### Steps to solve:

#### 1. **Identify the Coordinates of the Vertices**:
   - \( A(-2, 1) \)
   - \( B(0, 4) \)
   - \( C(4, 3) \)
   - \( D(2, -1) \)

#### 2. **Calculate the Length of Each Side**:
To find the perimeter, calculate the lengths of all the sides using the distance formula:
\[
\text{Distance between two points } (x_1, y_1) \text{ and } (x_2, y_2) \text{ is given by: }
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

- \( AB \): between points \( A(-2, 1) \) and \( B(0, 4) \)
- \( BC \): between points \( B(0, 4) \) and \( C(4, 3) \)
- \( CD \): between points \( C(4, 3) \) and \( D(2, -1) \)
- \( DA \): between points \( D(2, -1) \) and \( A(-2, 1) \)

#### 3. **Perimeter**:
Add the lengths of all sides to get the perimeter.

#### 4. **Calculate the Area**:
Since this is an irregular quadrilateral, you can use the **Shoelace Theorem** (or the surveyor's formula) to find the area of the polygon. The formula is:
\[
\text{Area} = \frac{1}{2} \left| x_1y_2 + x_2y_3 + \dots + x_ny_1 - (y_1x_2 + y_2x_3 + \dots + y_nx_1) \right|
\]
Where the points are listed in a cyclic order around the polygon.

I will now calculate the distances and the area.The calculations are as follows:

- **Perimeter of Quadrilateral \(ABCD\)**: \( \approx 16.67 \) units
- **Area of Quadrilateral \(ABCD\)**: \( 17.0 \) square units

Would you like further details or have any questions? 

Here are five follow-up questions to explore:

1. How can the Shoelace Theorem be applied to polygons with more than four vertices?
2. Can the same method be used to calculate the perimeter and area of a triangle?
3. How does the perimeter change if one of the vertices moves?
4. What are other methods to find the area of irregular polygons?
5. How would you calculate the area and perimeter of a circle or ellipse?

**Tip:** When calculating areas of complex shapes, breaking them down into simpler components, like triangles or rectangles, can often simplify the process.

Use the diagram to find the perimeter and the area of quadrilateral ABCD. The coordinates of the vertices are: A(-2,1), B(0,4), C(4,3), D(2,-1).

This problem involves finding the perimeter and area of a quadrilateral using coordinate geometry. By applying the distance formula for each side and using the Shoelace Theorem, students can calculate the area of irregular polygons. This method is suitable for students in Grades 8-10.

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