Question

Find the area of the following shape. You must show all work to receive credit.

shape with vertices located at negative 4 comma 0, 0 comma 4, 2 comma 2, negative 1 comma 2, and negative 1 comma 0

Answer 1

To find the area of the shape defined by the vertices \((-4, 0)\), \((0, 4)\), \((2, 2)\), \((-1, 2)\), and \((-1, 0)\), we can use the Shoelace theorem (or formula). This theorem provides a way to calculate the area of a polygon when the vertices are defined in the Cartesian plane.

### Step 1: List the vertices

We'll first list the vertices in a systematic order around the shape. To determine the order, we can visualize or sketch the points:

1. \((-4, 0)\)
2. \((-1, 0)\)
3. \((-1, 2)\)
4. \((2, 2)\)
5. \((0, 4)\)

### Step 2: Repeat the first vertex at the end

To apply the Shoelace theorem, we repeat the first vertex at the end of the list:

\[
\begin{array}{c}
(-4, 0) \\
(-1, 0) \\
(-1, 2) \\
(2, 2) \\
(0, 4) \\
(-4, 0) \\
\end{array}
\]

### Step 3: Apply the Shoelace theorem

The Shoelace theorem states that if you have vertices \((x_1, y_1)\), \((x_2, y_2)\), ..., \((x_n, y_n)\), the area \(A\) of the polygon can be computed using the formula:

\[
A = \frac{1}{2} \left| \sum_{i=1}^{n} (x_i y_{i+1} - y_i x_{i+1}) \right|
\]

Where \( (x_{n+1}, y_{n+1}) \) is the same as \( (x_1, y_1) \).

### Step 4: Set up the table for the calculations

We will now create two columns for our calculations:

\[
\begin{array}{c|c}
x & y \\
\hline
-4 & 0 \\
-1 & 0 \\
-1 & 2 \\
2 & 2 \\
0 & 4 \\
\end{array}
\]

And the corresponding \(y\) values shifted down 1 row:

\[
\begin{array}{c|c}
x & y \\
\hline
0 & -4 \\
0 & -1 \\
2 & -1 \\
2 & 2 \\
4 & 0 \\
\end{array}
\]

### Step 5: Compute the products

We compute the sums:

1. \(x_i y_{i+1}\):
- \((-4)(0) = 0\)
- \((-1)(0) = 0\)
- \((-1)(2) = -2\)
- \(2(4) = 8\)
- \(0(0) = 0\)

The total for \(x_i y_{i+1} = 0 + 0 - 2 + 8 + 0 = 6\)

2. \(y_i x_{i+1}\):
- \(0(-1) = 0\)
- \(0(-1) = 0\)
- \(2(2) = 4\)
- \(2(0) = 0\)
- \(4(-4) = -16\)

The total for \(y_i x_{i+1} = 0 + 0 + 4 + 0 - 16 = -12\)

### Step 6: Substitute into the area formula

Now substituting both sums into the area formula:

\[
A = \frac{1}{2} \left| 6 - (-12) \right| = \frac{1}{2} \left| 6 + 12 \right| = \frac{1}{2} \times 18 = 9
\]

Thus, the area of the shape is \(\boxed{9}\).