Let's solve this step-by-step by analyzing the polygon, which is decomposed into two triangles.

Step 1: Identify the Base and Height of Each Triangle

From the grid:

• The polygon is composed of two right triangles, and the dashed line in the middle represents the common base.
• Let's count the number of grid units:
  • Base (common to both triangles): The horizontal distance is 6 units (as it stretches from $x = 2$ to $x = 8$).
  • Height of the top triangle: The vertical distance from the base to the top vertex is 4 units.
  • Height of the bottom triangle: The vertical distance from the base to the bottom vertex is 3 units.

Step 2: Compute the Area of Each Triangle

The area of a triangle is given by: $\text{Area of a triangle} = \frac{1}{2} \times \text{base} \times \text{height}$

Top triangle:

$\text{Area of top triangle} = \frac{1}{2} \times 6 \times 4 = 12 \, \text{square units}$

Bottom triangle:

$\text{Area of bottom triangle} = \frac{1}{2} \times 6 \times 3 = 9 \, \text{square units}$

Step 3: Total Area of the Polygon

Now, we add the areas of both triangles: $\text{Total area} = 12 + 9 = 21 \, \text{square units}$

Thus, the total area of the shaded polygon is 21 square units.

Would you like more details or have any questions?

Here are 5 related questions to think about:

1. How would the area change if the height of the top triangle was doubled?
2. What if the base was 8 units instead of 6 units?
3. How can we verify that these triangles are right triangles using the grid?
4. How would the problem change if the triangles were not right triangles?
5. What if the polygon was divided into more than two triangles?

Tip: When calculating the area of composite shapes, breaking them into simpler shapes like triangles makes the calculation more manageable.