https://file.aiquickdraw.com/mathbot/file/1727892913931yv3e1s68.jpg

The diagram in the image shows a polygon with vertices labeled \( A \), \( B \), \( C \), \( D \), and \( E \). To find the area of the figure, I will break it into basic geometric shapes like rectangles and triangles.

The vertices' coordinates are approximately:
- \( A( -3, 4 ) \)
- \( B( 3, 4 ) \)
- \( C( 2, -1 ) \)
- \( D( 4, -7 ) \)
- \( E( -4, -7 ) \)

### Steps to calculate the area:
1. Divide the figure into two parts:
   - A rectangle: formed by \( A \), \( B \), \( E \), and \( D \).
   - A triangle: formed by \( B \), \( C \), and \( D \).

### Rectangle area:
The rectangle has vertices \( A( -3, 4 ) \), \( B( 3, 4 ) \), \( E( -4, -7 ) \), and \( D( 4, -7 ) \). Its height is the difference between the y-coordinates (4 and -7), and its width is the difference between the x-coordinates (3 and -3).

- Height: \( 4 - (-7) = 11 \)
- Width: \( 3 - (-3) = 6 \)

Area of rectangle = \( 6 \times 11 = 66 \) square units.

### Triangle area:
For the triangle formed by points \( B( 3, 4 ) \), \( C( 2, -1 ) \), and \( D( 4, -7 ) \), use the formula for the area of a triangle with given vertices:
\[
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\]
Substituting the coordinates:
\[
x_1 = 3, y_1 = 4 \quad x_2 = 2, y_2 = -1 \quad x_3 = 4, y_3 = -7
\]
\[
\text{Area} = \frac{1}{2} \left| 3(-1 - (-7)) + 2(-7 - 4) + 4(4 - (-1)) \right|
\]
\[
= \frac{1}{2} \left| 3(6) + 2(-11) + 4(5) \right| = \frac{1}{2} \left| 18 - 22 + 20 \right| = \frac{1}{2} \left| 16 \right| = 8
\]

### Total area:
The total area is the sum of the areas of the rectangle and the triangle:
\[
66 + 8 = 74 \text{ square units}
\]

Thus, the total area of the figure is **74 square units**.

Would you like further details or have any questions?

Here are 5 related questions to expand on:
1. How do you calculate the perimeter of this figure?
2. How would the area change if one of the vertices moved horizontally or vertically?
3. How does the formula for the area of a triangle change for different coordinate values?
4. What is the importance of breaking a complex shape into simpler shapes for area calculations?
5. Can you use another method (e.g., the Shoelace Theorem) to find the area of this polygon?

**Tip:** For shapes with irregular boundaries, breaking them into simpler geometric shapes is an efficient strategy to compute their area.

Find the area of the polygon with vertices A, B, C, D, and E.

This problem guides you through calculating the area of a polygon using its vertices in a coordinate plane. Learn to break the polygon into simple shapes like rectangles and triangles and use geometry formulas. Suitable for Grades 8-10.

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