Use the coordinate to compute the perimeter of a parallelogram whose vertices are located at the ordered pairs (0,0), (1,2), (2,1) and (1,-1).

To calculate the perimeter of a parallelogram with vertices at \((0,0)\), \((1,2)\), \((2,1)\), and \((1,-1)\), follow these steps:

### Step 1: Identify the vertices and the sides
The vertices are:
- \(A(0, 0)\)
- \(B(1, 2)\)
- \(C(2, 1)\)
- \(D(1, -1)\)

### Step 2: Calculate the lengths of the sides
We need to calculate the distance between consecutive vertices to find the lengths of the sides.

1. **Length of side AB**:
   \[
   AB = \sqrt{(1-0)^2 + (2-0)^2} = \sqrt{1 + 4} = \sqrt{5}
   \]

2. **Length of side BC**:
   \[
   BC = \sqrt{(2-1)^2 + (1-2)^2} = \sqrt{1 + 1} = \sqrt{2}
   \]

3. **Length of side CD**:
   \[
   CD = \sqrt{(1-2)^2 + (-1-1)^2} = \sqrt{1 + 4} = \sqrt{5}
   \]

4. **Length of side DA**:
   \[
   DA = \sqrt{(1-0)^2 + (-1-0)^2} = \sqrt{1 + 1} = \sqrt{2}
   \]

### Step 3: Calculate the perimeter
The perimeter \(P\) of the parallelogram is the sum of the lengths of all sides:
\[
P = AB + BC + CD + DA = \sqrt{5} + \sqrt{2} + \sqrt{5} + \sqrt{2} = 2\sqrt{5} + 2\sqrt{2}
\]

So, the perimeter of the parallelogram is \(2\sqrt{5} + 2\sqrt{2}\).

Would you like more details or have any questions?

Here are five related questions for further practice:
1. How would you find the area of this parallelogram?
2. What is the formula for the distance between two points in a coordinate plane?
3. How would the perimeter change if one of the vertices was moved?
4. How can you verify that the given vertices indeed form a parallelogram?
5. What is the importance of calculating the perimeter in real-world applications?

**Tip:** In coordinate geometry, the distance formula is a handy tool to find the lengths of sides and diagonals of polygons, helping to solve problems related to perimeters and areas.

Use the coordinates to compute the perimeter of a parallelogram whose vertices are located at the ordered pairs (0,0), (1,2), (2,1), and (1,-1).

Learn how to calculate the perimeter of a parallelogram given the vertices (0,0), (1,2), (2,1), and (1,-1). This problem involves the use of the distance formula and perimeter calculation. Perfect for students in Grades 8-10 studying coordinate geometry.

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