https://file.aiquickdraw.com/mathbot/file/1728696288970btdmcquy.jpg

Let's solve the problem step by step. The given shape is a polygon with vertices labeled \( A, B, C, D, E \), and the coordinates are:

- \( A(0, 0) \)
- \( B(2, 4) \)
- \( C(6, 2) \)
- \( D(4, -2) \)
- \( E(-2, -2) \)

### Step 1: Finding the Area
We will use the **Shoelace Theorem** (or surveyor's formula) for finding the area of a polygon with known coordinates. The formula is:

\[
\text{Area} = \frac{1}{2} \left| x_1y_2 + x_2y_3 + \cdots + x_ny_1 - (y_1x_2 + y_2x_3 + \cdots + y_nx_1) \right|
\]

Substituting the points \( A(0, 0), B(2, 4), C(6, 2), D(4, -2), E(-2, -2) \):

\[
\text{Area} = \frac{1}{2} \left| (0)(4) + (2)(2) + (6)(-2) + (4)(-2) + (-2)(0) - [(0)(2) + (4)(6) + (2)(4) + (-2)(-2) + (-2)(0)] \right|
\]

Calculating step by step:
- First part: \( (0)(4) + (2)(2) + (6)(-2) + (4)(-2) + (-2)(0) = 0 + 4 - 12 - 8 + 0 = -16 \)
- Second part: \( (0)(2) + (4)(6) + (2)(4) + (-2)(-2) + (-2)(0) = 0 + 24 + 8 + 4 + 0 = 36 \)

Now subtract and take the absolute value:

\[
\text{Area} = \frac{1}{2} \left| -16 - 36 \right| = \frac{1}{2} \times 52 = 26
\]

Thus, the area is **26 square units**.

### Step 2: Finding the Perimeter
The perimeter is the sum of the lengths of the sides. To find the length of each side, we will use the distance formula:

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

- **\( AB \)**:
\[
\text{Distance} = \sqrt{(2 - 0)^2 + (4 - 0)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}
\]
- **\( BC \)**:
\[
\text{Distance} = \sqrt{(6 - 2)^2 + (2 - 4)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}
\]
- **\( CD \)**:
\[
\text{Distance} = \sqrt{(4 - 6)^2 + (-2 - 2)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}
\]
- **\( DE \)**:
\[
\text{Distance} = \sqrt{(4 - (-2))^2 + (-2 - (-2))^2} = \sqrt{36 + 0} = 6
\]
- **\( EA \)**:
\[
\text{Distance} = \sqrt{(0 - (-2))^2 + (0 - (-2))^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]

Now, sum up all the lengths:

\[
\text{Perimeter} = 2\sqrt{5} + 2\sqrt{5} + 2\sqrt{5} + 6 + 2\sqrt{2}
\]

This simplifies to:

\[
\text{Perimeter} = 6\sqrt{5} + 6 + 2\sqrt{2}
\]

So, the perimeter is \( 6\sqrt{5} + 2\sqrt{2} + 6 \) units.

### Final Answer:
- **Area** = 26 square units
- **Perimeter** = \( 6\sqrt{5} + 2\sqrt{2} + 6 \) units

Let me know if you need any clarifications or further details!

Here are 5 questions to expand on this topic:
1. What is the Shoelace Theorem and how does it work for polygons?
2. How would you find the perimeter of a polygon if it had more sides?
3. Can the Shoelace Theorem be used for any type of polygon, and why?
4. What is the difference between exact values like \( \sqrt{5} \) and decimal approximations?
5. How would you approach calculating the area of a shape that is not on a grid?

**Tip**: When calculating the perimeter of polygons, always ensure to use the distance formula for each side, especially if the polygon is irregular or has diagonals.

Find the area and perimeter of the figure shown below. The polygon has vertices A(0, 0), B(2, 4), C(6, 2), D(4, -2), E(-2, -2).

Solve a problem where the area and perimeter of a polygon are calculated using the Shoelace Theorem and distance formula. The polygon has vertices A(0, 0), B(2, 4), C(6, 2), D(4, -2), E(-2, -2). Suitable for students in Grades 9-12.

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