Question

A (-1, -2) B (-2, 0) C (0, 2) D (2, 0) E (1, -2)
Use the given coordinates, what is the perimeter of the polygon? Round each calculation to the nearest tenth.

Answer 1

To find the perimeter of the polygon formed by the points A, B, C, D, and E, we need to calculate the distances between each pair of consecutive points, and then sum those distances.

The coordinates are:
- A (-1, -2)
- B (-2, 0)
- C (0, 2)
- D (2, 0)
- E (1, -2)

We will calculate the distance between each pair of points using the distance formula:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

### Step 1: Calculate distances

1. **Distance AB**:
\[
d_{AB} = \sqrt{(-2 - (-1))^2 + (0 - (-2))^2} = \sqrt{(-2 + 1)^2 + (0 + 2)^2} = \sqrt{(-1)^2 + (2)^2} = \sqrt{1 + 4} = \sqrt{5} \approx 2.2
\]

2. **Distance BC**:
\[
d_{BC} = \sqrt{(0 - (-2))^2 + (2 - 0)^2} = \sqrt{(0 + 2)^2 + (2)^2} = \sqrt{(2)^2 + (2)^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.8
\]

3. **Distance CD**:
\[
d_{CD} = \sqrt{(2 - 0)^2 + (0 - 2)^2} = \sqrt{(2)^2 + (0 - 2)^2} = \sqrt{(2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.8
\]

4. **Distance DE**:
\[
d_{DE} = \sqrt{(1 - 2)^2 + (-2 - 0)^2} = \sqrt{(-1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} \approx 2.2
\]

5. **Distance EA**:
\[
d_{EA} = \sqrt{(-1 - 1)^2 + (-2 - (-2))^2} = \sqrt{(-2)^2 + (0)^2} = \sqrt{4 + 0} = 2
\]

### Step 2: Calculate perimeter
Now, we can add all the distances together to find the perimeter:

\[
\text{Perimeter} = d_{AB} + d_{BC} + d_{CD} + d_{DE} + d_{EA}
\]
\[
\text{Perimeter} \approx 2.2 + 2.8 + 2.8 + 2.2 + 2 = 12
\]

### Final Answer
The perimeter of the polygon is approximately \( \boxed{12.0} \).