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Find the area of the shaded region above.

To find the area of the shaded region described in the problem, we need to break down the given shape and compute its area step-by-step. The shape is a polygon defined by the following vertices:

1. \((0, 0)\)
2. \((0, 13)\)
3. \((4, 21)\)
4. \((8, 13)\)
5. \((8, 0)\)
6. \((0, 0)\) (closing the shape)

The most effective way to find the area of this irregular polygon is to use the **Shoelace Theorem** (or Gauss's area formula for polygons). This theorem provides the area \( A \) of a polygon when the coordinates of its vertices are known.

### Shoelace Theorem Formula
For a polygon with vertices \((x_1, y_1)\), \((x_2, y_2)\), \(\ldots\), \((x_n, y_n)\) (listed in order), the area is given by:

\[
A = \frac{1}{2} \left| \sum_{i=1}^{n-1} (x_i \cdot y_{i+1} - y_i \cdot x_{i+1}) + (x_n \cdot y_1 - y_n \cdot x_1) \right|
\]

### Applying the Shoelace Theorem
Let’s use the coordinates of the vertices:

1. \((x_1, y_1) = (0, 0)\)
2. \((x_2, y_2) = (0, 13)\)
3. \((x_3, y_3) = (4, 21)\)
4. \((x_4, y_4) = (8, 13)\)
5. \((x_5, y_5) = (8, 0)\)

Substitute these into the formula:

\[
A = \frac{1}{2} \left| (0 \cdot 13 + 0 \cdot 21 + 4 \cdot 13 + 8 \cdot 0 + 8 \cdot 0) - (0 \cdot 0 + 13 \cdot 4 + 21 \cdot 8 + 13 \cdot 8 + 0 \cdot 0) \right|
\]

Simplifying further:

\[
A = \frac{1}{2} \left| 0 + 0 + 52 + 0 + 0 - (0 + 52 + 168 + 104 + 0) \right|
\]

\[
A = \frac{1}{2} \left| 52 - 324 \right|
\]

\[
A = \frac{1}{2} \times 272 = 136
\]

Thus, the area of the shaded region is:

\[
\boxed{136 \text{ square units}}
\]

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. How can the Shoelace Theorem be applied to a polygon with more vertices?
2. What other methods can be used to find the area of irregular polygons?
3. How does the Shoelace Theorem relate to the concept of determinants in linear algebra?
4. What is the formula for the perimeter of the given polygon?
5. How can we compute the centroid of this polygon?

**Tip:** When using the Shoelace Theorem, always list the vertices in a consistent clockwise or counterclockwise order to get the correct area value.

Find the area of the shaded region for the polygon defined by the points (0, 0), (0, 13), (4, 21), (8, 13), (8, 0).

Learn how to find the area of a polygon using the Shoelace Theorem. This problem involves applying coordinate geometry to a shape defined by the points (0, 0), (0, 13), (4, 21), (8, 13), and (8, 0). Suitable for Grades 9-11 students looking to understand polygon area calculation.

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